I will summarize some recent developments in the study of
quantum entanglement spectra.  I will also discuss work
performed in collaboration with R. Thomale and A. Bernevig
on entanglement spectra in spin chains.  Typically,
bipartite entanglement entropy and spectra have been studied
in the case of spatial partitions, i.e. A denotes the left
half of a spin chain and B the right half, and the
eigenvalues of the reduced density matrix of the A component
comprise the entanglement spectrum (ES). We find that for
the spin-half Heisenberg model that a remarkable structure
in the ES is revealed if the partition is performed in
momentum space, i.e. A = left-movers and B = right-movers. 
 Further classifying the entanglement eigenstates by total
crystal momentum, we observe a universal low-lying portion
of the ES with specific multiplicities separated from a
higher-lying nonuniversal set of levels by an entanglement
gap, similar to what was observed by Li and Haldane (2008)
for the fractional quantum Hall effect.  Indeed, the
momentum space ES for the Heisenberg chain is understood in
terms of the proximity of the Haldane-Shastry model, which
corresponds to a fixed point with no nonuniversal
corrections, and whose ground state is related to the
Laughlin state.   We further explore the behavior of the ES
as one tunes through the dimerization transition in a model
with next-nearest-neighbor exchange.