  We consider a bipartite scenario where two parties hold ensembles of
$1/2$-spins which can only be measured collectively. We give numerical
arguments supporting the conjecture that in this scenario no Bell inequality
can be violated for arbitrary numbers of spins if only first order moment
observables are available. We then give a recipe to achieve a significant Bell
violation with a split many-body system when this restriction is lifted. This
highlights the strong requirements needed to detect bipartite quantum
correlations in many-body systems device-independently.
