  A recent experiment reported the first violation of a Bell correlation
witness in a many-body system [Science 352, 441 (2016)]. Following discussions
in this paper, we address here the question of the statistics required to
witness Bell correlated states, i.e. states violating a Bell inequality, in
such experiments. We start by deriving multipartite Bell inequalities involving
an arbitrary number of measurement settings, two outcomes per party and one-
and two-body correlators only. Based on these inequalities, we then build up
improved witnesses able to detect Bell-correlated states in many-body systems
using two collective measurements only. These witnesses can potentially detect
Bell correlations in states with an arbitrarily low amount of spin squeezing.
We then establish an upper bound on the statistics needed to convincingly
conclude that a measured state is Bell-correlated.
