The ubiquitous variability of quasars across a wide range of wavelengths and timescales encodes critical information about the structure and dynamics of the circumnuclear emitting regions that are too small to be directly resolved, as well as detailed underlying physics of accretion and feedback processes in these active supermassive black holes. We emphasize the importance of studying quasar variability with time-domain spectroscopy, focusing on two science cases: (1) reverberation mapping (RM) to measure the broad-line region sizes and black hole masses in distant quasars; (2) spectroscopic follow-up of extreme variability quasars that dramatically change their continuum and broad-line flux within several years. We highlight the need for dedicated optical-infrared spectroscopic survey facilities in the coming decades to accompany wide-area time-domain imaging surveys, including: (1) the next phase of the Sloan Digital Sky Survey (SDSS-V; ~2020-2025), an all-sky, time-domain multi-object spectroscopic survey with 2.5m-class telescopes; (2) the planned Maunakea Spectroscopic Explorer, a dedicated 10m-class spectroscopic survey telescope with a 1.5 sq. deg field-of-view and multiplex of thousands of fibers in both optical and near-IR (J+H) to begin operations in 2029; (3) the Time-domain Spectroscopic Observatory (TSO), a proposed Probe-class ~1.3m telescope at L2, with imaging and spectroscopy (R=200, 1800) in 4 bands (0.3 - 5 micron) and rapid slew capability to 90% of sky, which will extend the coverage of Hbeta to z=8.
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