  In recent years, the use of information principles to understand quantum
correlations has been very successful. Unfortunately, all principles considered
so far have a bipartite formulation, but intrinsically multipartite principles,
yet to be discovered, are necessary for reproducing quantum correlations. Here,
we introduce local orthogonality, an intrinsically multipartite principle
stating that events involving different outcomes of the same local measurement
must be exclusive, or orthogonal. We prove that it is equivalent to
no-signaling in the bipartite scenario but more restrictive for more than two
parties. By exploiting this non-equivalence, it is then demonstrated that some
bipartite supra-quantum correlations do violate local orthogonality when
distributed among several parties. Finally, we show how its multipartite
character allows revealing the non-quantumness of correlations for which any
bipartite principle fails. We believe that local orthogonality is a crucial
ingredient for understanding no-signaling and quantum correlations.
