  I give a brief review of higher dimensional quantum Hall effect (QHE) and how
one can use a general framework to describe the lowest Landau level dynamics as
a noncommutative field theory whose semiclassical limit leads to anomaly free
bulk-edge effective actions in any dimension. I then present the case of QHE on
complex projective spaces and focus on the entanglement entropy for integer QHE
in even spatial dimensions. In the case of $\nu=1$, a semiclassical analysis
shows that the entanglement entropy is proportional to the phase-space area of
the entangling surface with a universal overall constant, same for any
dimension as well as abelian or nonabelian background magnetic fields. This is
modified for higher Landau levels.
