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Abstract The measured ages of massive, quiescent galaxies atz∼ 3–4 imply that massive galaxies quench as early asz∼ 6. While the number of spectroscopic confirmations of quiescent galaxies atz< 3 has increased over the years, there are only a handful atz> 3.5. We report spectroscopic redshifts of one secure (z= 3.757) and two tentative (z= 3.336 andz= 4.673) massive ( $\log \left(M_{*}/M_{\odot }\right)>10.3$) quiescent galaxies with 11 hr of Keck/MOSFIREK-band observations. Our candidates were selected from the FLAMINGOS-2 Extragalactic Near-InfraredK-band Split (FENIKS) survey, which uses deep Gemini/Flamingos-2K_bK_rimaging optimized for increased sensitivity to the characteristic red colors of galaxies atz> 3 with a strong Balmer/4000 Å break. The rest-frameUVJand (ugi)_scolors of three out of four quiescent candidates are consistent with 1–2 Gyr old stellar populations. This places these galaxies as the oldest objects at these redshifts, and challenges the notion that quiescent galaxies atz> 3 are all recently quenched, post-starburst galaxies. Our spectroscopy shows that the other quiescent-galaxy candidate is a broad-line active galactic nucleus (z= 3.594) with strong, redshifted Hβ+ [OIII] emission with a velocity offset > 1000 km s⁻¹, indicative of a powerful outflow. The star formation history of our highest redshift candidate suggests that its progenitor was already in place byz∼ 7–11, reaching ∼10¹¹M_⊙byz≃ 8. These observations reveal the limit of what is possible with deep near-infrared photometry and targeted spectroscopy from the ground and demonstrate that secure spectroscopic confirmation of quiescent galaxies atz> 4 is feasible only with JWST.