Many properties of galaxies depend on the environment they are located in. At z < 1, this environmental dependence is most clearly seen in galaxy clusters. At high redshifts, galaxy clusters are split into a number of dark halos (DHs) and unbound regions. In this study, among those structures, we focus on the most massive DHs, or ""proto-cluster cores”. We search for proto-cluster cores at cosmic noon in the COSMOS field, and examine the properties of galaxies in them. According to the extended Press-Schechter theory, DHs with M_DH > ~10^13 Msun at z~2 can grow into present-day clusters. Using the COSMOS2015 galaxy catalog by Laigle+16, we identify ~150 core candidates. Clustering analysis shows that these candidates are indeed hosted by DHs with the targeted mass range. We find that the member galaxies of these cores have a much higher passive fraction than field galaxies. We also find that the stellar mass function of member galaxies is more top-heavy than that of field galaxies.