Bell showed 50 years ago that quantum theory is non-local via his celebrated inequalities, turning the issue of quantum non-locality from a matter of taste into a matter of test. Years later, Hardy proposed a test for non-locality without inequality, which is a kind of ‘something-versus-nothing’ argument. Hardyʼs test for n particles induces an n-partite Bellʼs inequality with two dichotomic local measurements for each observer, which has been shown to be violated by all entangled pure states. Our first result is to show that the Bell–Hardy inequality arising form Hardyʼs non-locality test is tight for an arbitrary number of parties, i.e., it defines a facet of the Bell polytope in the given scenario. On the other hand quantum theory is not that non-local since it forbids signaling and even not as non-local as allowed by non-signaling conditions, i.e., quantum mechanical predictions form a strict subset of the so called non-signaling polytope. In the scenario of each observer measuring two dichotomic observables, Fritz established a duality between the Bell polytope and the non-signaling polytope: tight Bellʼs inequalities, the facets of the Bell polytope, are in a one-to-one correspondence with extremal non-signaling boxes, the vertices of the non-signaling polytope. Our second result is to provide an alternative and more direct formula for this duality. As an example, the tight Bell–Hardy inequality gives rise to an extremal non-signaling box that serves as a natural multipartite generalization of Popescu–Rohrlich box.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.