It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely non-local, then any quantum mechanical predictions can be recovered. In this paper, we consider general hidden variable models which can have both local and non-local parts. We then show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a non-trivial local part, such as the model proposed by Leggett.
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