Universal speed limit
In the late 19’th century it was noticed that the speed of light is independent of the reference frame. For example, the speed of light emitted from a fast moving train is the same as the speed of light emitted from a lamp post at the train station. In modern parlance, we observe that the speed of light emitted from a fast moving free electron, say it is moving at a speed very close to the speed of light in a beam tube at CERN, is the same as the speed of light emitted from a local electron bound to a an atom inside the control room at CERN. This is very strange! This is very counter-intuitive but it is a fact. It makes more sense to us now because we know that light consists of quanta called photons. It is easier for us to imagine that once the photon comes into existence it moves at the speed of light. It does not matter how fast the photon emitting object was moving. Once it releases the photon the photon moves at a constant speed. But, this was not so easy to intuit in the late 19’th century because nobody knew that light consisted of quanta.
Hendrik Lorentz expressed these observations mathematically long before Einstein investigated the other consequences. The combined efforts of Hendrik Lorentz and Albert Einstein resulted in a theory known as the Special Relativity. Einstein was given a Nobel prize for his proposal that light consists of quanta (photons) but his contributions to Special Relativity as well as the General Relativity – his theory of gravitation – was ignored by the Nobel committee.
Einstein took a bold step and theorized that the constancy of the speed of photons implies that there is a universal speed limit for all objects. According to Einstein, all objects, not just the photons, obey this law. Not just photons and other material objects…the law also applies to any messaging (information exchange). According to the theory of relativity (Special Relativity) no object, no force, or no information can move faster than the speed of light in vacuum.
Hendrik Lorentz was a great physicist but he was not bold enough to suggest that his own formula applied to all objects.
Special Relativity theory has been tested in great detail over the years. You may remember the latest test involving neutrinos at the OPERA experiment at CERN. In September 2011, OPERA researchers reported that muon neutrinos traveled faster than light in their experiment. This was a very unusual announcement because experimental physicists do not rush to make announcements like this. Experimental physicists are very conservative people. As a matter of fact, not all members of the OPERA collaboration participated in that announcement. They were right because in March 2012, OPERA collaboration announced that they have found a loose fibre optic cable connecting a GPS receiver to an electronic card in a computer. That fault was responsible for the erroneous conclusion. They retracted their previous claim and announced that muon neutrinos do not travel faster than light. So, once again, the theory of relativity has been proven to be correct.
Tension not conflict
It is important to state clearly at the outset that all measurements in quantum mechanical settings are consistent with the theory of Special Relativity. Physicists have not been able to find a single example where signals or elementary particles travel faster than light in vacuum. So, what is the point of this article?
The title of this article uses the word “tension” instead of “conflict.” There is no conflict between relativity and quantum mechanics (QM) as far as measurements are concerned. But, there is tension because QM concepts are not consistent with the universal speed law.
In order to demonstrate the tension between relativity and QM most physicists mention quantum entanglement but the tension is clearly evident in other areas of QM as well. I would start with the wavefunction collapse, the superposition principle and the uncertainty principle. You may want to take a look at my post titled “QM wavefunction and its many interpretations” for the background information.
Wavefunction collapse is not consistent with relativity
According to the Copenhagen interpretation of QM a system is completely described by a wavefunction which evolves smoothly in time, except when a measurement is made, at which point it instantaneously collapses to one of the possible states (eigenstate) of the observable that is measured. In the Copenhagen interpretation wavefunction collapse is a fundamental, a priori principle.
According to relativity nothing happens instantaneously. Instantaneous wavefunction collapse is not consistent with relativity.
Quantum superposition is not consistent with relativity
According to the quantum superposition principle a system can exist in many states simultaneously. This means that an elementary particle can be in many places at the same time. Quantum supersposition is not consistent with relativity because relativity says that a particle cannot move from one place to another instantaneously. The claim that a particle can be in many places simultaneously implies movement faster than speed of light.
Uncertainty principle is not consistent with relativity
According to QM, the more precisely the position of a particle is determined, the less precisely its momentum – a quantity related to its velocity and mass – can be known, and vice versa. If you locate the particle you will not know its momentum precisely and if you measure its momentum you will not know its location precisely. This phenomenon is known as the Heisenberg’s uncertainty principle. Another aspect of the Heisenberg uncertainty principle is that the energy can fluctuate wildly over a small interval of time.
It is important to emphasize the fact that quantum mechanical uncertainty is not statistical uncertainty (sample average, standard deviation, etc). It is also important to note that quantum mechanical uncertainty is not related to measurement errors either. The Heisenberg uncertainty principle points to an intrinsic uncertainty. In my post titled “Why is uncertainty intrinsic in the Cosmos” I tried to explain this with the universal principle that Consciousness seeks freedom from bondage.
So, when we locate the particle precisely its momentum – a quantity related to speed – will be uncertain and therefore there is a theoretical possibility that the speed of particle exceeds the speed of light in vacuum. This has never been observed but it is a theoretical possibility according to QM. This possibility is enhanced by the fact that energy can fluctuate within a small interval of time.
The Quantum Field Theory (QFT) comes up with a conceptual solution to this problem and explains that QM and relativity are consistent. I will say more about the QFT resolution below.
QFT saves the day!
Quantum Field Theory (QFT) makes QM consistent with relativity by replacing one mystery with another. QFT replaces the QM wavefunction with the “quantum field.” QM wavefunction does not create or destroy particles. The “quantum field” does.
Lisa Randall explains the quantum field as follows:
“Quantum field theory, the tool with which we study particles, is based on eternal, omnipresent objects that can create and destroy those particles. These objects are the ‘fields’ of quantum field theory. Like the classical electromagnetic fields that inspired their name, quantum fields are objects that permeate spacetime. But quantum fields play a different role. They create or absorb elementary particles. According to quantum field theory, particles can be produced or destroyed anywhere and anytime.” 
According to QFT there is no motion per se. In fact, QFT denies that there are elementary particles. According to QFT a particle is just a ripple on the quantum field. The “ripple” appears to be moving but it is not motion in the classical sense. Rather, it is the appearance and disappearance of certain attributes (particle properties). In the QFT language particles are created and destroyed by the quantum field. For example, there is an electron field…it can create and destroy electrons anywhere in space-time. This circumvents the problem of wavefunction collapse but as I said earlier this is just a conceptual trick. We replaced the mystery of QM wavefunction with another mystery: quantum field.
This resolution is similar in spirit to the ingenious way the Big Bang and the Cosmic Inflation theories circumvent the problem of the universal speed limit law. During the early phases of the Big Bang and especially during the super fast expansion of the universe known as the Cosmic Inflation particles were moving apart at speeds much faster than the universal speed limit. Big Bang cosmologists say that there is no conflict with relativity because space itself was stretching. QFT resolution is similar. QFT says that the universal speed limit law does not apply to the quantum field because the quantum field permeates the entire space-time.
I know it is just a trick and I can live with it. Who am I to argue? QFT is the most successful and most useful theory out there. What bothers me about QFT is that there is a separate field for each particle type. I registered my complaint in the “Why so many fields?” article.
Spin singlet state (quantum entanglement)
It is possible to prepare an electron-positron pair occupying a quantum state called a spin singlet. The singlet state can be thought of as a composite particle that has spin=0. Electron and positron of this singlet will have magnetic moments equal in magnitude but opposite in sign. The electron-positron pair prepared in a spin singlet state is said to be entangled. According to QM the entangled pair will act as if it is a single particle even when they are separated miles apart.
While they are separated we can measure the magnetic moment of one of them relative to an external magnetic field line in any direction. The other particle of the pair will have the opposite magnetic moment along the same direction.
So how does the other particle of the pair instantaneously know what magnetic field line direction was chosen in the experiment miles away? No information can move faster than the speed of light in vacuum. According to relativity instantaneous communication is not possible. Some scientists would comment on this by saying that the question is irrelevant because the QM wavefunction is in a spin singlet state therefore the pair is acting as if it is a single (composite) particle. This is a very good answer but the mystery does not go away. The mystery is the spin singlet state itself. It is clear that the QM wavefunction is a non-local construct and the wavefunction collapse is a non-local phenomenon. The non-locality of QM is not consistent with relativity.
QM is not consistent with space-time concept
“In modern quantum physics, entanglement is fundamental; furthermore, space is irrelevant—at least in quantum information science, space plays no central role and time is a mere discrete clock parameter. In relativity, space-time is fundamental and there is no place for nonlocal correlations. To put the tension in other words: No story in space-time can tell us how nonlocal correlations happen; hence, nonlocal quantum correlations seem to emerge, somehow, from outside space-time.” 
The treatment of time in Quantum Mechanics is very problematic. In other theories of physics we model the time evolution of the system and hope that measurements (snapshots in time) conform to the model. Quantum Mechanics works in the opposite direction. Quantum Mechanics models the measurement process. The dynamics (time evolution of the system) is derived in an awkward way.
Measurement is local but it may have non-local effects according to Quantum Mechanics.
 Lisa Randall, “Warped Passages”, Harper Perennial (2005)
 N. Gisin, “Quantum Nonlocality: How Does Nature Do It?” (Science, Vol. 326 No. 5958, pp. 1357 – 1358)