  Entanglement is usually quantified by von Neumann entropy, but its properties
are much more complex than what can be expressed with a single number. We show
that the three distinct dynamical phases known as thermalization, Anderson
localization, and many-body localization are marked by different patterns of
the spectrum of the reduced density matrix for a state evolved after a quantum
quench. While the entanglement spectrum displays Poisson statistics for the
case of Anderson localization, it displays universal Wigner-Dyson statistics
for both the cases of many-body localization and thermalization, albeit the
universal distribution is asymptotically reached within very different time
scales in these two cases. We further show that the complexity of entanglement,
revealed by the possibility of disentangling the state through a
Metropolis-like algorithm, is signaled by whether the entanglement spectrum
level spacing is Poisson or Wigner-Dyson distributed.
