  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.
