  Bell's theorem proves that some quantum state correlations can only be
explained by bipartite non-classical resources. The notion of genuinely
multipartite nonlocality (GMNL) was later introduced to conceptualize the fact
that nonclassical resources involving more than two parties in a nontrivial way
may be needed to account for some quantum correlations. In this letter, we
first recall the contradictions inherent to the historical definition of GMNL.
Second, we turn to one of its redefinitions, called
Local-Operations-and-Shared-Randomness GMNL (LOSR-GMNL), proving that all
caterpillar graph states (including cluster states) have this second property.
Finally, we conceptualize a third, alternative definition, which we call
Local-Operations-and-Neighbour-Communication GMNL (LONC-GMNL), that is adapted
to situations in which short-range communication between some parties might
occur. We show that cluster states do not have this third property, while GHZ
states do. Beyond its technical content, our letter illustrates that rigorous
conceptual work is needed before applying the concepts of genuinely
multipartite nonlocality, genuine multipartite entanglement or entanglement
depth to benchmark the nonclassicality of some experimentally-produced quantum
system. We note that most experimental works still use witnesses based on the
historical definitions of these notions, which fail to reject models based on
bipartite resources.
