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There is no action at a distance in quantum mechanics, that is a laymen's misconception. If there was, it would not be compatible with special relativity, but it is compatible as they are already unified under the framework of quantum field theory. The No-communication theorem is a rather simple proof that shows there is no "sharing at a distance" in quantum mechanics. It is an entirely local theory. The misconception arises from people misinterpreting Bell's theorem which says quantum mechanics is not compatible with a local hidden variable theory, so people falsely conclude it's a nonlocal theory, but this is just false because quantum mechanics is not a hidden variable theory, and so it is not incompatible with locality. It is a local theory. Bell's theorem only shows it is nonlocal if you introduce hidden variables, meaning the theorem is really only applicable to a potential replacement to quantum mechanics and is not even applicable to quantum mechanics itself. It is applicable to things like pilot wave theory, but not to quantum theory.