  The measurement of quantum entanglement in many-body systems remains
challenging. One experimentally relevant fact about quantum entanglement is
that in systems whose degrees of freedom map to free fermions with conserved
total particle number, exact relations hold relating the Full Counting
Statistics associated with the bipartite charge fluctuations and the sequence
of R\' enyi entropies. We draw a correspondence between the bipartite charge
fluctuations and the entanglement spectrum, mediated by the R\' enyi entropies.
In the case of the integer quantum Hall effect, we show that it is possible to
reproduce the generic features of the entanglement spectrum from a measurement
of the second charge cumulant only. Additionally, asking whether it is possible
to extend the free fermion result to the $\nu=1/3$ fractional quantum Hall
case, we provide numerical evidence that the answer is negative in general. We
further address the problem of quantum Hall edge states described by a
Luttinger liquid, and derive expressions for the spectral functions of the real
space entanglement spectrum at a quantum point contact realized in a quantum
Hall sample.
