A galactic core-collapse supernova is a powerful neutrino source of which the signals can be picked up by a water Cherenkov detector on the Earth. From an astrophysical point of view, the signals reveal the dynamics of core-collapse supernova explosion and the subsequent cooling of a proto-neutron star (PNS). In this regard, we compare the neutrino emission profiles from the recent 1D hydrodynamics simulation by Mirizzi et al. (2016) with the historical SN1987A data through a statistical goodness-of-fit test. Such test reveals the tension between the data and rapid PNS cooling prescribed by the convection treatment employed in the simulation. The implications will be discussed. From a quantum-mechanical point of view, on the other hand, the supernova neutrino flux is so intensive such that a huge degree of wave-packet overlap is estimated. Such overlap may give rise to an interference effect known as the Hanbury Brown and Twiss (HBT) effect. We derive the solution for a 3D Gaussian wave packet and, with such solution, the joint-detection probability. We demonstrate that an observable interference occurs if the joint-detection were to render the two detected neutrinos in the same phase space cell. Upon further examination, however, we conclude that such effect is difficult to observe from neutrinos in practical experimental settings.