While proving that no matter can reach the speed of light (a fact which I call the kinetic energy barrier), Einstein uses the fact that he can calculate the velocity and position of an electron. However, if quantum effects apply, then it seems to create a problem in Einstein's assumptions themselves. How is the proof of the kinetic energy barrier true even though quantum effects exist in nature?
It is well established by now that the speed of light barrier does not apply to quantum particles, and this property makes the construction of relativistic quantum field theories and other relativistic quantum systems tightly constrained. The argument that you can't transmit signals faster than light is fine, but particles are not necessarily signals, because if you make a particle over here, and measure a particle over there, you might not be measuring the particle you created, but another identical particle you created from the vacuum.
So relativistic field theory, with its faster-than-light particles, requires that there are no unique particles, that all particles have identical copies. Further, the faster than light motion can be back-in-time in different frames, and back-in-time motion means that every particle must have a back-in-time partner, called its antiparticle.
The quantum field restores locality. So even though quantum particles can propagate faster than light, information can't propagate faster than light. The quantum fields are the quantities which tell you what information you can gain locally by experiments. The particles in a Hamiltonian formulation are nonlocally related to the quantum fields (but the two are related more simply in a particle path integral formulation).
The proof that quantum particles cannot be restricted to less than light speed is simple: the restriction that the energy is positive means that the frequency is positive, while the restriction to forward propagation inside the light cone means that the propagator vanishes outside the future lightcone, so in particular, into the past. and there is no function which vanishes in a half-plane whose Fourier transform also vanishes in a big unbounded region like this. This is covered in this answer: the causality and the anti-particles
Because no one's figured out a way to use and/or validate it, but see Has anyone tried transmitting a signal with equally-tuned Casimir plates across the quantum vacuum?.