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

Aspects of higher dimensional quantum Hall effect: Bosonization, entanglement entropy