Nonlocality is arguably one of the most fundamental and counterintuitive aspects of quantum theory. Nonlocal correlations could, however, be even more nonlocal than quantum theory allows, while still complying with basic physical principles such as no-signaling. So why is quantum mechanics not as nonlocal as it could be? Are there other physical or information-theoretic principles which prohibit this? So far, the proposed answers to this question have been only partially successful, partly because they are lacking genuinely multipartite formulations. In Nat. Comm. 4, 2263 (2013) we introduced the principle of Local Orthogonality (LO), an intrinsically multipartite principle which is satisfied by quantum mechanics but is violated by non-physical correlations. Here we further explore the LO principle, presenting new results and explaining some of its subtleties. In particular, we show that the set of no-signaling boxes satisfying LO is closed under wirings, present a classification of all LO inequalities in certain scenarios, show that all extremal tripartite boxes with two binary measurements per party violate LO, and explain the connection between LO inequalities and unextendible product bases.
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