  It has been recently realized that strong interactions in topological Bloch
bands give rise to the appearance of novel states of matter. Here we study
connections between these systems -- fractional Chern insulators and the
fractional quantum Hall states -- via generalization of a gauge-fixed
Wannier-Qi construction in the cylinder geometry. Our setup offers a number of
important advantages compared to the earlier exact diagonalization studies on a
torus. Most notably, it gives access to edge states and to a single-cut orbital
entanglement spectrum, hence to the physics of bulk-edge correspondence. It is
also readily implemented in the state-of-the-art density matrix renormalisation
group method that allows for numerical simulations of significantly larger
systems. We demonstrate our general approach on examples of flat-band models on
ruby and kagome lattices at bosonic filling fractions $\nu=1/2$ and $\nu=1$,
which show the signatures of (non)-Abelian phases, and establish the
correspondence between the physics of edge states and the entanglement in the
bulk. Notably, we find that the non-Abelian $\nu=1$ phase can be stabilized by
purely on-site interactions in the presence of a confining potential.
