Simulating the motion of atoms and molecules is a challenging problem, especially when the dynamics of the atomic nuclei need to be treated quantum mechanically. In this thesis we analyse three path-integral based approximations for computing quantum time-correlation functions: constant-uncertainty molecular dynamics (CUMD), the fitted harmonic approximation (FHA) and windowed centroid molecular dynamics (WCMD). The CUMD method has been proposed as a simple and efficient method to incorporate nuclear quantum effects in molecular simulations. The method applies a position-momentum constraint between system replicas based on the uncertainty principle. After reproducing the results from the original publication, we show that the method uses an ad hoc fix to apply the constraint which makes it impractical when extended to systems larger than toy models. The FHA is proposed in this work as a locally harmonic approximation to the linearised-semi classical initial value representation. We find that the FHA results in time correlation functions which are similar to and in some cases better than those obtained from the local gaussian approximation (LGA). In its current implementation, the FHA method is a proof of principle which has been applied to test systems in one and two dimensions, but the results obtained are sufficiently promising to suggest that future implementations of the FHA could compete with the LGA. The WCMD method is proposed in this work as a simple method for removing contributions from delocalised ring polymers in centroid molecular dynamics calculations, in which they are known to cause artificial red shifts in vibrational spectra. We apply the WCMD method to two dimensional test systems and find that by filtering out the delocalised ring polymers we are able to eliminate the artificial red shift. This result is extremely promising and suggests that the WCMD method should be extendable in future work to treat systems such as gas phase and liquid water.