  To identify which principles characterize quantum correlations, it is
essential to understand in which sense this set of correlations differs from
that of almost quantum correlations. We solve this problem by invoking the
so-called no-restriction hypothesis, an explicit and natural axiom in many
reconstructions of quantum theory stating that the set of possible measurements
is the dual of the set of states. We prove that, contrary to quantum
correlations, no generalised probabilistic theory satisfying the no-restriction
hypothesis is able to reproduce the set of almost quantum correlations.
Therefore, any theory whose correlations are exactly, or very close to, the
almost quantum correlations necessarily requires a rule limiting the possible
measurements. Our results suggest that the no-restriction hypothesis may play a
fundamental role in singling out the set of quantum correlations among other
non-signalling ones.
