  We use particle entanglement spectra to characterize bosonic quantum Hall
states on lattices, motivated by recent studies of bosonic atoms on optical
lattices. Unlike for the related problem of fractional Chern insulators, very
good trial wavefunctions are known for fractional quantum Hall states on
lattices. We focus on the entanglement spectra for the Laughlin state at
$\nu=1/2$ for the non-Abelian Moore-Read state at $\nu=1$. We undertake a
comparative study of these trial states to the corresponding groundstates of
repulsive two-body or three-body contact interactions on the lattice. The
magnitude of the entanglement gap is studied as a function of the interaction
strength on the lattice, giving insights into the nature of Landau-level
mixing. In addition, we compare the performance of the entanglement gap and
overlaps with trial wavefunctions as possible indicators for the topological
order in the system. We discuss how the entanglement spectra allow to detect
competing phases such as a Bose-Einstein condensate.
