EPR paradox
In 1935 Einstein, Podolsky, and Rosen published a thought experiment meant to show that quantum mechanics, however successful, must be an incomplete description of reality. They imagined two particles that interact and then fly apart, leaving their positions and momenta correlated. By measuring one particle you could predict, with certainty, a property of the distant other without ever touching it — so, they argued, that property must have been real all along.
Their reasoning rested on two assumptions they treated as obvious. Locality: what you choose to measure here cannot instantly affect physical reality over there. Realism: if you can predict a quantity with certainty without disturbing a system, that quantity corresponds to something genuinely present. Quantum mechanics says the distant particle has no definite value until measured, which to EPR seemed to leave a gap that 'hidden' details should fill.
Crucially, EPR did not claim quantum mechanics was wrong, only that it left out part of the story. For decades this looked like a matter of taste or philosophy. The deep surprise came thirty years later, when John Bell showed the EPR assumptions make testable predictions — and experiments have since sided with quantum mechanics, ruling out the kind of locally-real completion Einstein had hoped for.
Perfect prediction at a distance led EPR to argue a 'complete' theory must add hidden elements of reality.
EPR is often called a paradox, but it is really an argument, and history judged its premises rather than its logic. Its locality-plus-realism assumptions were the part later experiments showed nature does not obey.