  We employ the recent results on the generalization of the $c$-theorem to
2+1-d to derive non-perturbative results for strongly interacting quantum field
theories, including QED-3 and the critical theory corresponding to certain
quantum phase transitions in condensed matter systems. In particular, by
demanding that the universal constant part of the entanglement entropy
decreases along the renormalization group flow ("F-theorem"), we find bounds on
the number of flavors of fermions required for the stability of QED-3 against
chiral symmetry breaking and confinement. In this context, the exact results
known for the entanglement of superconformal field theories turn out to be
quite useful. Furthermore, the universal number corresponding to the ratio of
the entanglement entropy of a free Dirac fermion to that of free scalar plays
an interesting role in the bounds derived. Using similar ideas, we also derive
strong constraints on the nature of quantum critical points in condensed matter
systems with "topological order".
