  Spin squeezing inequalities (SSI) represent a major tool to probe quantum
entanglement among a collection of few-level atoms, and are based on collective
spin measurements and their fluctuations. Yet, for atomic ensembles of spin-$j$
atoms and ultracold spinor gases, many experiments can image the populations in
all Zeeman sublevels $s=-j, -j+1, \dots, j$, potentially revealing finer
features of quantum entanglement not captured by SSI. Here we present a
systematic approach which exploits Zeeman-sublevel population measurements in
order to construct novel entanglement criteria, and illustrate our approach on
ground states of spin-1 and spin-2 Bose-Einstein condensates. Beyond these
specific examples, our approach allows one to infer, in a systematic manner,
the optimal permutationally-invariant entanglement witness for any given set of
collective measurements in an ensemble of $d$-level quantum systems.
