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We report two secure ($z=3.775, 4.012$) and one tentative ($z\approx3.767$)spectroscopic confirmations of massive and quiescent galaxies through $K$-bandobservations with Keck/MOSFIRE and VLT/X-Shooter. The stellar continuumemission, the absence of strong nebular emission lines and the lack ofsignificant far-infrared detections confirm the passive nature of theseobjects, disfavoring the alternative solution of low-redshift dustystar-forming interlopers. We derive stellar masses of$\mathrm{log}(M_{\star}/M_\odot)\sim11$ and ongoing star formation ratesplacing these galaxies $\gtrsim 1-2$ dex below the main sequence at theirredshifts. The adopted parametrization of the star formation history suggeststhat these sources experienced a strong ($\langle \rm SFR \rangle \sim1200-3500\,M_\odot\,\mathrm{yr}^{-1}$) and short ($\sim 50$ Myr) burst of starformation, peaking $\sim 150-500$ Myr before the time of observation, allproperties reminiscent of the characteristics of sub-millimeter galaxies (SMGs)at $z>4$. We investigate this connection by comparing the comoving numberdensities and the properties of these two populations. We find a fair agreementonly with the deepest sub-mm surveys detecting not only the most extremestarbursts, but also more normal galaxies. We support these findings by furtherexploring the Illustris-TNG cosmological simulation, retrieving populations ofboth fully quenched massive galaxies at $z\sim3-4$ and SMGs at $z\sim4-5$, withnumber densities and properties in agreement with the observations at $z\sim3$,but in increasing tension at higher redshift. Nevertheless, as suggested by theobservations, not all the progenitors of quiescent galaxies at these redshiftsshine as bright SMGs in their past and, similarly, not all bright SMGs quenchby $z\sim3$, both fractions depending on the threshold assumed to define theSMGs themselves.

Quiescent galaxies 1.5 billion years after the Big Bang and their progenitors