  Based on quantum-mechanical path-integral molecular dynamics simulations the
impact of nuclear quantum effects on the vibrational and hydrogen bond dynamics
in liquid water is investigated. The instantaneous fluctuations in the
frequencies of the O-H stretch modes are calculated using the wavelet method of
time series analysis, while the time scales of the vibrational spectral
diffusion are determined from frequency-time correlation functions, joint
probability distributions, as well as the slope of three-pulse photon echo. We
find that the inclusion of nuclear quantum effects leads not only to a redshift
of the vibrational frequency distribution by around 130~cm$^{-1}$, but also to
an acceleration of the vibrational dynamics by as much as 30$\%$. In addition,
quantum fluctuations also entail a significantly faster decay of correlation in
the initial diffusive regime, which is agreement with recent vibrational echo
experiments.
