A proto-cluster core is the most massive dark matter halo (DMH) in a given proto-cluster. To reveal the galaxy formation in core regions, we search for proto-cluster cores at $zsim 2$ in $sim 1.5, mathrm{deg}^{2}$ of the COSMOS field. Using pairs of massive galaxies ($log(M_{*}/M_{odot})geq11$) as tracers of cores, we find 75 candidate cores, among which 54% are estimated to be real. A clustering analysis finds that these cores have an average DMH mass of $2.6_{-0.8}^{+0.9}times 10^{13}, M_{odot}$, or $4.0_{-1.5}^{+1.8}, times 10^{13} M_{odot}$ after contamination correction. The extended Press-Schechter model shows that their descendant mass at $z=0$ is consistent with Fornax-like or Virgo-like clusters. Moreover, using the IllustrisTNG simulation, we confirm that pairs of massive galaxies are good tracers of DMHs massive enough to be regarded as proto-cluster cores. We then derive the stellar mass function (SMF) and the quiescent fraction for member galaxies of the 75 candidate cores. We find that the core galaxies have a more top-heavy SMF than field galaxies at the same redshift, showing an excess at $log(M_{*}/M_{odot})gtrsim 10.5$. The quiescent fraction, $0.17_{-0.04}^{+0.04}$ in the mass range $9.0leq log(M_{*}/M_{odot})leq 11.0$, is about three times higher than that of field counterparts, giving an environmental quenching efficiency of $0.13_{-0.04}^{+0.04}$. These results suggest that stellar mass assembly and quenching are accelerated as early as at $zsim 2$ in proto-cluster cores.
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