  We have applied path integral molecular dynamics simulations to investigate
nuclear quantum effects on the vibrational dynamics of water molecules at the
water-air interface. 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 vibrational spectral diffusion are
determined from frequency-time correlation functions and joint probability
distributions, as well as the hydrogen bond number correlation functions. We
find that the inclusion of nuclear quantum effects leads not only to a redshift
in the vibrational frequency distribution by about 120~cm$^{-1}$ for both the
bulk and interfacial water molecules, but also to an acceleration of the
vibrational dynamics on the water-air interface by as much as 60$\%$. In
addition, a blueshift of about 30 ~cm$^{-1}$ is seen in the vibrational
frequency distribution of interfacial water molecules compared to that of the
bulk. Furthermore, the dynamics of water molecules beyond the topmost two
molecular layers was found to be essentially similar to that of bulk water.
