An instantaneous normal mode (INM) theory is presented for quantum time correlation functions. It is argued that the INM formalism for classical correlation functions is particularly amenable to quantum correction. The intermolecular (Rayleigh) and allowed vibrational Raman spectra of liquid CS2 are calculated as an illustration. The Applequist-Quicksall polarizability model is employed, yielding the correct values for both the molecular polarizability and its derivatives with respect to the normal coordinates, Agreement with experiment is reasonable for the intermolecular Raman and for some aspects of the allowed Raman, but not for the linewidth. A brief discussion is given regarding the future developments which will be needed for an accurate INM theory of vibrational line shapes.