  We show that the model wave functions used to describe the fractional quantum
Hall effect have exact representations as matrix product states (MPS). These
MPS can be implemented numerically in the orbital basis of both finite and
infinite cylinders, which provides an efficient way of calculating arbitrary
observables. We extend this approach to the charged excitations and numerically
compute their Berry phases. Finally, we present an algorithm for numerically
computing the real-space entanglement spectrum starting from an arbitrary
orbital basis MPS, which allows us to study the scaling properties of the
real-space entanglement spectra on infinite cylinders. The real-space
entanglement spectrum obeys a scaling form dictated by the edge conformal field
theory, allowing us to accurately extract the two entanglement velocities of
the Moore-Read state. In contrast, the orbital space spectrum is observed to
scale according to a complex set of power laws that rule out a similar
collapse.
