
This formulation of the Einstein – Podolsky – Rosen paradox, was published in Physics Review in March 1935: Can Quantum Mechanical reality be considered to be complete?
The first formulation: From Heisenberg’s Uncertainty Principle we cannot simultaneously measure to exact precision the position and momentum of a quantum mechanical system. Therefore if the wavefunction is an eigenfunction of the position operator, it cannot be an eigenfunction of the momentum operator and vice versa. Suppose we have a system consisting of 2 identical particles, A and B. After interaction, they are separated far apart from each other without undergoing any other interactions. And since this is a “thought experiment”, we can let them travel until they become spacelike separated, i.e. anything which happens to one particle cannot interfere with the other (interference would require signals traveling above light speed, which were forbidden by Einstein's concepts of locality and separability). Suppose we measure the momentum of A exactly, then the wavefunction would be an eigenfunction of momentum for A. From the Law of Conservation of Momentum, the momentum of B would be of equal magnitude but opposite direction. And now, let us measure the position of B (at the same time as the measurement of momentum of A). The wavefunction would then collapse and we would get an eigenvalue for its position. This gives rise to a contradiction! From Heisenberg’s Uncertainty Principle, we cannot know the momentum and position of a particle at a certain time to infinite precision. Therefore, the Copenhagen Interpretation of Quantum Mechanics is incomplete!

These images obtained from Soshichi Uchii (2004) 
Special relativity tells us that nothing
can travel faster than the speed of light. By making use of this fact, a quite
similar “thought experiment” was devised. However, this assumption
of locality will come back to haunt Einstein in Bohr's
reply.
But the 2 particles are assumed to
be light years apart and thus spacelike. This instantaneous collapse of the
wavefunction of B requires nonlocality or in Einstein’s words: “spooky
action at a distance”.
Therefore, the Copenhagen interpretation of Quantum Mechanics violates Special
Theory of Relativity, and thus is incomplete.