Recent work of L. Polterovich exibits a link between rigidity properties in symplectic topology and some properties of quantum observables (positive operator valued measures). This link is established through Berezin-Toeplitz quantization. In view of this, it would be desirable to extend the asymptotic results on Berezin-Toeplitz quantization to symbols with lower regularity. In this talk, I will discuss some recent results in this direction as well as some open questions.