Topological quantum computation (and related matters in condensed matter theory) stimulates a number of complexity-theoretic questions concerning TQFT invariants, especially for 3-dimensional manifolds. I’ll begin by reviewing some of these and the motivations. Then, as an easy application of a major result of Cai and Chen [J. ACM, 2017], I’ll prove the following: for any fixed 3-manifold invariant of either Reshetikhin-Turaev or Turaev-Viro-Barrett-Westbury type, there is either a polynomial time algorithm to compute the invariant (on all triangulated 3-manifolds) or it is #P-hard. I’ll conclude by opining on the ways one might turn this “in principle” dichotomy into an “effective” dichotomy. This is joint work with Nicolas Bridges.