  Bell inequalities are natural tools that allow one to certify the presence of
nonlocality in quantum systems. The known constructions of multipartite Bell
inequalities contain, however, correlation functions involving all observers,
making their experimental implementation difficult. The main purpose of this
work is to explore the possibility of witnessing nonlocality in multipartite
quantum states from the easiest-to-measure quantities, that is, the two-body
correlations. In particular, we determine all three and four-partite Bell
inequalities constructed from one and two-body expectation values that obey
translational symmetry, and show that they reveal nonlocality in multipartite
states. Also, by providing a particular example of a five-partite Bell
inequality, we show that nonlocality can be detected from two-body correlators
involving only nearest neighbours. Finally, we demonstrate that any
translationally invariant Bell inequality can be maximally violated by a
translationally invariant state and the same set of observables at all sites.
We provide a numerical algorithm allowing one to seek for maximal violation of
a translationally invariant Bell inequality.
