We report new Chandra observations of seven optically faint, z ~ 4 radio-quiet quasars (RQQs). We have combined these new observations with previous Chandra observations of RQQs to create a sample of 174 sources. These sources have 0.1 < z < 4.7, and 1044 ergs s-1 < νLν (2500 Å) < 1048 ergs s-1. The X-ray detection fraction is 90%. We find that the X-ray loudness of RQQs decreases with UV luminosity and increases with redshift. The model that is best supported by the data has a linear dependence of optical to X-ray ratio αox on cosmic time and a quadratic dependence of αox on log LUV, where αox becomes X-ray quiet more rapidly at higher log LUV. We find no significant evidence for a relationship between the X-ray photon index Γx and the UV luminosity, and we find marginally significant evidence that the X-ray continuum flattens with increasing z (2 σ). The Γx-z anticorrelation may be the result of X-ray spectral curvature, redshifting of a Compton reflection component into the observed Chandra band, and/or redshifting of a soft excess out of the observed Chandra band. Using the results for Γx, we show that the αox-z relationship is unlikely to be a spurious result caused by redshifting of the observable X-ray spectral region. A correlation between αox and z implies evolution of the accretion process. We present a qualitative comparison of these new results with models for accretion disk emission.