  We describe the asymptotic behaviour of the quantum propagator generated by a
Berezin-Toeplitz operator with real-valued principal symbol. We also give
precise asymptotics for smoothed spectral projectors associated with the
operator in the autonomous case; this leads us to introducting quantum states
associated with immersed Lagrangian submanifolds. These descriptions involve
geometric quantities of two origins, coming from lifts of the Hamiltonian flow
to the prequantum bundle and the canonical bundle respectively. The latter are
the main contribution of this article and are connected to the Maslov indices
appearing in trace formulas, as will be explained in a forthcoming paper.
