We study the time lags between the continuum emission of quasars at different wavelengths, based on more than four years of multi-band (g, r, i, z) light curves in the Pan-STARRS Medium Deep Fields. As photons from different bands emerge from different radial ranges in the accretion disk, the lags constrain the sizes of the accretion disks. We select 240 quasars with redshifts of z 1 or z 0.3 that are relatively emission-line free. The light curves are sampled from day to month timescales, which makes it possible to detect lags on the scale of the light crossing time of the accretion disks. With the code JAVELIN, we detect typical lags of several days in the rest frame between the g band and the riz bands. The detected lags are ∼2-3 times larger than the light crossing time estimated from the standard thin disk model, consistent with the recently measured lag in NGC 5548 and microlensing measurements of quasars. The lags in our sample are found to increase with increasing luminosity. Furthermore, the increase in lags going from g − r to g − i and then to g − z is slower than predicted in the thin disk model, particularly for high-luminosity quasars. The radial temperature profile in the disk must be different from what is assumed. We also find evidence that the lags decrease with increasing line ratios between ultraviolet Fe ii lines and Mg ii, which may point to changes in the accretion disk structure at higher metallicity.