The Witten-Reshetikhin-Turaev Topological Quantum Field Theory in particular provides us with the so-called quantum representations of mapping class groups. The geometric construction of these involves geometric quantization of moduli spaces, which produced a holomorphic vector bundle over Teichm\"uller space. This bundle supports a projectively flat connection constructed by algebraic geometric techniques by Hitchin. We will present a a Toeplitz operator approximation formula for the parallel transport of the Hitchin connection. We will discuss applications of this