We report two secure ( ) and one tentative (z 3.767) spectroscopic confirmations of massive and quiescent galaxies through K-band observations with Keck/MOSFIRE and Very Large Telescope/X-Shooter. The stellar continuum emission, absence of strong nebular emission lines, and lack of significant far-infrared detections confirm the passive nature of these objects, disfavoring the alternative solution of low-redshift dusty star-forming interlopers. We derive stellar masses of log(M /M ) ∼ 11 and ongoing star formation rates placing these galaxies 1-2 dex below the main sequence at their redshifts. The adopted parameterization of the star formation history suggests that these sources experienced a strong ( M yr−1) and short (∼50 Myr) burst of star formation, peaking ∼150-500 Myr before the time of observation, all properties reminiscent of the characteristics of submillimeter galaxies (SMGs) at z > 4. We investigate this connection by comparing the comoving number densities and the properties of these two populations. We find a fair agreement only with the deepest submillimeter surveys detecting not only the most extreme starbursts but also more normal galaxies. We support these findings by further exploring the Illustris TNG cosmological simulation, retrieving populations of both fully quenched massive galaxies at z ∼ 3-4 and SMGs at z ∼ 4−5, with number densities and properties in agreement with the observations at z ∼ 3 but in increasing tension at higher redshift. Nevertheless, as suggested by the observations, not all of the progenitors of quiescent galaxies at these redshifts shine as bright SMGs in their past, and, similarly, not all bright SMGs quench by z ∼ 3, both fractions depending on the threshold assumed to define the SMGs themselves.