  Local measurements acting on entangled quantum states give rise to a rich
correlation structure in the multipartite scenario. We introduce a versatile
technique to build families of Bell inequalities witnessing different notions
of multipartite nonlocality for any number of parties. The idea behind our
method is simple: a known Bell inequality satisfying certain constraints, for
example the Clauser-Horne-Shimony-Holt inequality, serves as the $seed$ to
build new families of inequalities for more parties. The constructed
inequalities have a clear operational meaning, capturing an essential feature
of multipartite correlations: their violation implies that numerous subgroups
of parties violate the inequality chosen as seed. The more multipartite
nonlocal the correlations, the more subgroups can violate the seed. We
illustrate our construction using different seeds and designing Bell
inequalities to detect $m$-way nonlocal multipartite correlations, in
particular, $genuine$ $multipartite$ $nonlocal$ correlations -- the strongest
notion of multipartite nonlocality. For one of our inequalities we prove
analytically that a large class of pure states that are genuine multipartite
entangled exhibit genuine multipartite nonlocality for any number of parties,
even for some states that are almost product. We also provide numerical
evidence that this family is violated by all genuine multipartite entangled
pure states of three and four qubits. Our results make us conjecture that this
family of Bell inequalities can be used to prove the equivalence between
genuine multipartite pure-state entanglement and nonlocality for any number of
parties.
