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In this talk we describe our recent work on discrete quantum theory based onGalois fields. In particular, we discuss how discrete quantum theory sheds newlight on the foundations of quantum theory and we review an explicit model ofsuper-quantum correlations we have constructed in this context. We also discussthe larger questions of the origins and foundations of quantum theory, as wellas the relevance of super-quantum theory for the quantum theory of gravity.

Quantum Systems based upon Galois Fields: from Sub-quantum to Super-quantum Correlations