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Local measurements acting on entangled quantum states give rise to a richcorrelation structure in the multipartite scenario. We introduce a versatiletechnique to build families of Bell inequalities witnessing different notionsof multipartite nonlocality for any number of parties. The idea behind ourmethod is simple: a known Bell inequality satisfying certain constraints, forexample the Clauser-Horne-Shimony-Holt inequality, serves as the $seed$ tobuild new families of inequalities for more parties. The constructedinequalities have a clear operational meaning, capturing an essential featureof multipartite correlations: their violation implies that numerous subgroupsof parties violate the inequality chosen as seed. The more multipartitenonlocal the correlations, the more subgroups can violate the seed. Weillustrate our construction using different seeds and designing Bellinequalities to detect $m$-way nonlocal multipartite correlations, inparticular, $genuine$ $multipartite$ $nonlocal$ correlations -- the strongestnotion of multipartite nonlocality. For one of our inequalities we proveanalytically that a large class of pure states that are genuine multipartiteentangled exhibit genuine multipartite nonlocality for any number of parties,even for some states that are almost product. We also provide numericalevidence that this family is violated by all genuine multipartite entangledpure states of three and four qubits. Our results make us conjecture that thisfamily of Bell inequalities can be used to prove the equivalence betweengenuine multipartite pure-state entanglement and nonlocality for any number ofparties.

A versatile construction of Bell inequalities for the multipartite scenario