The classical and quantum dynamics of an excess proton in water is studied by molecular dynamics simulations. The electronic structure of the system is described by an extended multistate valence-bond Hamiltonian that allows for the breaking and formation of O-H+ bonds. The proton quantum character is treated by means of an effective (path-integral) proton-transfer surface. Whereas classical simulations predict that the hydrated proton appears in a mixture of H5O2+ and H9O4+ structures, inclusion of proton quantization leads to the prevalence of H5O2+. The proton-transfer mechanism can be described mostly as the translocation of a transient H5O2+ structure across the water hydrogen-bond network. The computed lifetime of a particular H5O2+ is close to 2 ps, a value compatible with experimental estimates.