  The characterization of the set of quantum correlations in Bell scenarios is
a problem of paramount importance for both the foundations of quantum mechanics
and quantum information processing in the device-independent scenario. However,
a clear-cut (physical or mathematical) characterization of this set remains
elusive and many of its properties are still unknown. We provide here simple
and general analytical conditions that are necessary for an arbitrary bipartite
behaviour to be quantum. Although the conditions are not sufficient, we
illustrate the strength and non-triviality of these conditions with a few
examples. Moreover, we provide several applications of this result: we prove a
quantitative separation of the quantum set from extremal nonlocal no-signaling
behaviours in several general scenarios, we provide a relation to obtain
Tsirelson bounds for arbitrary Bell inequalities and a construction of Bell
expressions whose maximal quantum value is attained by a maximally entangled
state of any given dimension.
