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The measured ages of massive, quiescent galaxies at z∼3-4 imply that massive galaxies quench as early as z~6. While the number of spectroscopic confirmations of quiescent galaxies at z<3 has increased over the years, there are only a handful at z> 3.5. We report spectroscopic redshifts of one secure (z=3.757) and two tentative (z=3.336 and z=4.673) massive quiescent galaxies with 11 hr of Keck/MOSFIRE K-band observations. Our candidates were selected from the FLAMINGOS-2 Extragalactic Near-Infrared K-band Split (FENIKS) survey, which uses deep Gemini/Flamingos-2 Kb/Kr imaging optimized for increased sensitivity to the characteristic red colors of galaxies at z>3 with a strong Balmer/4000 Å break. The rest-frame UVJ and ugi colors 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 at z>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 Hbeta+[OIII] emission with a velocity offset >1000 km/s, indicative of a powerful outflow. The star formation history of our highest redshift candidate suggests that its progenitor was already in place by z~7-11, reaching ∼10^11 M⊙ by z~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 at z>4 is feasible only with JWST.