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The assumption of free will - the ability of an experimentalist to makerandom choices - is central to proving the indeterminism of quantum resources,the primary tool in quantum cryptography. Relaxing the assumption in a Belltest allows violation of the usual classical threshold by correlating therandom number generators used to select measurements with the devices thatperform them. In this paper, we examine not only these correlations, but thoseacross multiple runs of the experiment. This enables an explicit exposition ofthe optimal cheating strategy and how the correlations manifest themselveswithin this strategy. Similar to other recent results, we prove that thereremain Bell violations for a sufficiently high, yet non-maximal degree of freewill which cannot be simulated by a classical attack, regardless of how manyruns of the experiment those choices are correlated over.

Limited measurement dependence in multiple runs of a Bell test