  In this paper we investigate the problem of constructing Topological Quantum
Field Theories (TQFTs) to quantize algebraic invariants. We exhibit necessary
conditions for quantizability based on Euler characteristics. In the case of
surfaces, also provide a partial answer in terms of sufficient conditions by
means of almost-TQFTs and almost-Frobenius algebras for wide TQFTs. As an
application, we show that the Poincar\'e polynomial of $G$-representation
varieties is not a quantizable invariant by means of a monoidal TQFTs for any
algebraic group $G$ of positive dimension.
