  We report two secure ($z=3.775, 4.012$) and one tentative ($z\approx3.767$)
spectroscopic confirmations of massive and quiescent galaxies through $K$-band
observations with Keck/MOSFIRE and VLT/X-Shooter. The stellar continuum
emission, the absence of strong nebular emission lines and the 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
$\mathrm{log}(M_{\star}/M_\odot)\sim11$ and ongoing star formation rates
placing these galaxies $\gtrsim 1-2$ dex below the main sequence at their
redshifts. The adopted parametrization of the star formation history suggests
that these sources experienced a strong ($\langle \rm SFR \rangle \sim
1200-3500\,M_\odot\,\mathrm{yr}^{-1}$) and short ($\sim 50$ Myr) burst of star
formation, peaking $\sim 150-500$ Myr before the time of observation, all
properties reminiscent of the characteristics of sub-millimeter 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 sub-mm 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 $z\sim3-4$ and SMGs at $z\sim4-5$, with
number densities and properties in agreement with the observations at $z\sim3$,
but in increasing tension at higher redshift. Nevertheless, as suggested by the
observations, not all the progenitors of quiescent galaxies at these redshifts
shine as bright SMGs in their past and, similarly, not all bright SMGs quench
by $z\sim3$, both fractions depending on the threshold assumed to define the
SMGs themselves.
