Context. The Juno probe that currently orbits Jupiter measures its gravitational moments with great accuracy. Preliminary results suggest that the core of the planet may be eroded. While great attention has been paid to the material properties of elements constituting the envelope, little is known about those that constitute the core. This situation clutters our interpretation the Juno data and modeling of giant planets and exoplanets in general. Aims. We calculate the high-pressure melting temperatures of three potential components of the cores of giant planets, water, iron, and a simple silicate, MgSiO3, to investigate the state of the deep inner core. Methods. We used ab initio molecular dynamics simulations to calculate the high-pressure melting temperatures of the three potential core components. The planetary adiabats were obtained by solving the hydrostatic equations in a three-layer model adjusted to reproduce the measured gravitational moments. Recently developed ab initio equations of state were used for the envelope and the core. Results. We find that the cores of the giant and ice-giant planets of the solar system differ because the pressure–temperature conditions encountered in each object correspond to different regions of the phase diagrams. For Jupiter and Saturn, the results are compatible with a diffuse core and mixing of a significant fraction of metallic elements in the envelope, leading to a convective and/or a double-diffusion regime. We also find that their solid cores vary in nature and size throughout the lifetimes of these planets. The solid cores of the two giant planets are not primordial and nucleate and grow as the planets cool. We estimate that the solid core of Jupiter is 3 Gyr old and that of Saturn is 1.5 Gyr old. The situation is less extreme for Uranus and Neptune, whose cores are only partially melted. Conclusions. To model Jupiter, the time evolution of the interior structure of the giant planets and exoplanets in general, their luminosity, and the evolution of the tidal effects over their lifetimes, the core should be considered as crystallizing and growing rather than gradually mixing into the envelope due to the solubility of its components.