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

Translationally invariant multipartite Bell inequalities involving only two-body correlators