  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 $z\sim 2$ 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 $9.0\leq \log(M_{*}/M_{\odot})\leq
11.0$, 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
$z\sim 2$ in proto-cluster cores.
