[Abridged] Using public data from the NMBS and CANDELS surveys, we study the population of massive galaxies at z>3 to identify the potential progenitors of z~2 compact, massive, quiescent (CMQ) galaxies, furthering our understanding of the evolution of massive galaxies. Our work is enabled by high-resolution CANDELS images and accurate photometric redshifts, stellar masses and star formation rates (SFRs) from 37-band NMBS photometry. The total number of z>3 massive galaxies is consistent with the number of massive quiescent (MQ) galaxies at z~2, implying that the SFRs for all of these galaxies must be much lower by z~2. We discover 4 CMQ galaxies at z>3, pushing back the time for which such galaxies have been observed. However, the volume density for these galaxies is significantly less than that of galaxies at z<2 with similar masses, SFRs, and sizes, implying that additional CMQ galaxies must be created in the ~1 Gyr between z=3 and z=2. We find 5 star-forming galaxies at z~3 that are compact (Re<1.4 kpc) and have stellar mass M*>10^(10.6)Msun, likely to become members of the CMQ galaxy population at z~2. We evolve the stellar masses and SFRs of each individual z>3 galaxy adopting 5 different star formation histories (SFHs) and studying the resulting population of massive galaxies at z=2.3. We find that declining or truncated SFHs are necessary to match the observed number density of MQ galaxies at z~2, whereas a constant SFH results in a number density significantly smaller than observed. All of our assumed SFHs imply number densities of CMQ galaxies at z~2 that are consistent with the observed number density. Better agreement with the observed number density of CMQ galaxies at z~2 is obtained if merging is included in the analysis and better still if star formation quenching is assumed to shortly follow the merging event, as implied by recent models of formation of MQ galaxies.
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