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As an all-sky survey, NASA's $TESS$ mission is able to detect the brightestand rarest types of transiting planetary systems, including young planets thatenable study of the evolutionary processes that occur within the first billionyears. Here, we report the discovery of a young, multi-planet system orbitingthe bright K4.5V star, TOI-712 ($V = 10.838$, $M_\star =0.733_{-0.025}^{+0.026} M_\odot$, $R_\star = 0.674\pm0.016 R_\odot$, $T_{\rmeff} = 4622_{-60}^{+61}$ K). From the $TESS$ light curve, we measure a rotationperiod of 12.48 days, and derive an age between about $500$ Myr and 1.1 Gyr.The photometric observations reveal three transiting mini-Neptunes ($R_b =2.049^{+0.12}_{-0.080} R_\oplus$, $R_c = 2.701^{+0.092}_{-0.082} R_\oplus$,$R_d = 2.474^{+0.090}_{-0.082} R_\oplus $), with orbital periods of $P_b =9.531$ days, $P_c = 51.699$ days, and $P_d = 84.839$ days. After modeling thethree-planet system, an additional Earth-sized candidate is identified,TOI-712.05 ($P = 4.32$ days, $R_P = 0.81 \pm 0.11 R_\oplus$). We calculate thatthe habitable zone falls between 0.339 and 0.844 au (82.7 and 325.3 days),placing TOI-712 d near its inner edge. Among planetary systems harboringtemperate planets, TOI-712 ($T = 9.9$) stands out as a relatively young starbright enough to motivate further characterization.

TOI-712: a system of adolescent mini-Neptunes extending to the habitable zone