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This Letter examines how the sizes, structures, and color gradients of galaxies change along the quiescent sequence. Our sample consists of ~400 quiescent galaxies at $1.0\le z\le2.5$ and $10.1 \le \log{M_*/M_\odot}\le11.6$ in three CANDELS fields. We exploit deep multi-band HST imaging to derive accurate mass profiles and color gradients, then use an empirical calibration from rest-frame UVJ colors to estimate galaxy ages. We find that -- contrary to previous results -- the youngest quiescent galaxies are not significantly smaller than older quiescent galaxies at fixed stellar mass. These `post-starburst' galaxies only appear smaller in half-light radii because they have systematically flatter color gradients. The strength of color gradients in quiescent galaxies is a clear function of age, with older galaxies exhibiting stronger negative color gradients (i.e., redder centers). Furthermore, we find that the central mass surface density $Σ_1$ is independent of age at fixed stellar mass, and only weakly depends on redshift. This finding implies that the central mass profiles of quiescent galaxies do not significantly change with age; however, we find that older quiescent galaxies have additional mass at large radii. Our results support the idea that building a massive core is a necessary requirement for quenching beyond $z=1$, and indicate that post-starburst galaxies are the result of a rapid quenching process that requires structural change. Furthermore, our observed color gradient and mass profile evolution supports a scenario where quiescent galaxies grow inside-out via minor mergers.