  We consider the trial wavefunctions for the Fractional Quantum Hall Effect
(FQHE) that are given by conformal blocks, and construct their associated edge
excited states in full generality. The inner products between these edge states
are computed in the thermodynamic limit, assuming generalized screening (i.e.
short-range correlations only) inside the quantum Hall droplet, and using the
language of boundary conformal field theory (boundary CFT). These inner
products take universal values in this limit: they are equal to the
corresponding inner products in the bulk 2d chiral CFT which underlies the
trial wavefunction. This is a bulk/edge correspondence; it shows the equality
between equal-time correlators along the edge and the correlators of the bulk
CFT up to a Wick rotation. This approach is then used to analyze the
entanglement spectrum (ES) of the ground state obtained with a bipartition
A\cupB in real-space. Starting from our universal result for inner products in
the thermodynamic limit, we tackle corrections to scaling using standard
field-theoretic and renormalization group arguments. We prove that generalized
screening implies that the entanglement Hamiltonian H_E = - log {\rho}_A is
isospectral to an operator that is local along the cut between A and B. We also
show that a similar analysis can be carried out for particle partition. We
discuss the close analogy between the formalism of trial wavefunctions given by
conformal blocks and Tensor Product States, for which results analogous to ours
have appeared recently. Finally, the edge theory and entanglement spectrum of
px + ipy paired superfluids are treated in a similar fashion in the appendix.
