  We theoretically examine entanglement in fractional quantum hall states,
explicitly taking into account and emphasizing the quasi-two-dimensional nature
of experimental quantum Hall systems. In particular, we study the entanglement
entropy and the entanglement spectrum as a function of the finite layer
thickness $d$ of the quasi-two-dimensional system for a number of filling
fractions $\nu$ in the lowest and the second Landau levels: $\nu$ = 1/3, 7/3,
1/2, and 5/2. We observe that the entanglement measures are dependent on which
Landau level the electrons fractionally occupy, and find that filling factions
1/3 and 7/3, which are considered to be Laughlin states, weaken with $d$ in the
lowest Landau level ($\nu$=1/3) and strengthen with $d$ in the second Landau
level ($\nu$=7/3). For the enigmatic even-denominator $\nu=5/2$ state, we find
that entanglement in the ground state is consistent with that of the
non-Abelian Moore-Read Pfaffian state at an optimal thickness $d$. We also find
that the single-layer $\nu = 1/2$ system is not a fractional quantum Hall state
consistent with the experimental observation. In general, our theoretical
findings based on entanglement considerations are completely consistent with
the results based on wavefunction overlap calculations.
