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Bell's theorem renders quantum correlations distinct from those of anylocal-realistic model. Although being stronger than classical correlations,quantum correlations are limited by the Tsirelson bound. This bound, however,applies for Hermitian, commutative operators corresponding to non-signalingobservables in Alice's and Bob's spacelike-separated labs. As an attempt toexplore theories beyond quantum mechanics and analyze the uniqueness of thelatter, we examine in this work the extent of non-local correlations whenrelaxing these fundamental assumptions, which allows for theories withnon-local signaling. We prove that, somewhat surprisingly, the Tsirelson boundin the Bell-Clauser-Horne-Shimony-Holt scenario, and similarly other relatedbounds on non-local correlations, remain effective as long as we maintain theHilbert space structure of the theory. Furthermore, in the case of Hermitianobservables we find novel relations between non-locality, local correlations,and signaling. We demonstrate that such non-local signaling theories arenaturally simulated by quantum systems of parafermionic zero modes. Wenumerically study the derived bounds in parafermionic systems, confirming thebounds' validity yet finding a drastic difference between correlations of'signaling' and 'non-signaling' sets of observables. We also propose anexperimental procedure for measuring the relevant correlations.

Bounds on nonlocal correlations in the presence of signaling and their application to topological zero modes