  Bell's theorem was a cornerstone for our understanding of quantum theory, and
the establishment of Bell non-locality played a crucial role in the development
of quantum information. Recently, its extension to complex networks has been
attracting a growing attention, but a deep characterization of quantum
behaviour is still missing for this novel context. In this work we analyze
quantum correlations arising in the bilocality scenario, that is a tripartite
quantum network where the correlations between the parties are mediated by two
independent sources of states. First, we prove that non-bilocal correlations
witnessed through a Bell-state measurement in the central node of the network
form a subset of those obtainable by means of a separable measurement. This
leads us to derive the maximal violation of the bilocality inequality that can
be achieved by arbitrary two-qubit quantum states and arbitrary projective
separable measurements. We then analyze in details the relation between the
violation of the bilocality inequality and the CHSH inequality. Finally, we
show how our method can be extended to n-locality scenario consisting of n
two-qubit quantum states distributed among n+1 nodes of a star-shaped network.
