Random quasi-phase-matching (RQPM) is a nonlinear optical phenomenon known to occurnaturally in polycrystalline materials. When driven by a pulsed source in the infrared, RQPM in ZnSe/S can generate second-harmonic light. This second-harmonic generation (SHG) can reach optical powers comparable to those of conventional ultrafast sources, on the order of hundreds of milliwatts. These pulses are not well characterized, nor has a complete analysis on them been done to determine factors such as compressibility. Previous measurements of the light generated by RQPM have focused on the power spectra and have said nothing about the spectral phase, which is required for making predictions about the temporal profile and coherence of the light. Here, we use frequency-resolved optical gating (FROG) to reconstruct a temporal description of a laser pulse produced by SHG via RQPM in ZnS. The experiment is based on a collinear SHG beam geometry that can be easily reconfigured for additional measurements, including linear and interferometric autocorrelations. After measuring the pulse, we perform an additional measurement in order to resolve an ambiguity inherent to SHG FROG. Finally, we will report on an experiment that suggests pulses produced by RQPM in ZnS might be compressible to durations of 35 fs or shorter.