An important invariant of a path-connected topological space X is the number of homomorphisms from the fundamental group of X to a finite, non-abelian, simple group G. Kuperberg and Samperton proved that, although these invariants can be powerful, they are often computationally intractable, particularly when X is an integral homology 3-sphere. More specifically, they prove that the problem of counting such homomorphisms is #P-complete via a reduction from a known #P-complete circuit satisfiability problem. Their model constructs X from a well-chosen Heegaard surface and a mapping class in its Torelli group. We will introduce the basics of complexity for counting problems, summarize the reduction used by K-S to bound the problem of counting homomorphisms, and discuss some of the topological and quantum computing implications of their results.