  We address the formation of massive stellar spheroids between redshifts $z=4$
and 1 using a suite of AMR hydro-cosmological simulations. The spheroids form
as bulges, and the spheroid mass growth is partly driven by violent disc
instability (VDI) and partly by mergers. A kinematic decomposition to disc and
spheroid yields that the mass fraction in the spheroid is between 50\% and 90\%
and is roughly constant in time, consistent with a cosmological steady state of
VDI discs that are continuously fed from the cosmic web. The density profile of
the spheroid is typically "classical", with a Sersic index $n = 4.5\pm 1$,
independent of whether it grew by mergers or VDI and independent of the
feedback strength. The disc is characterized by $n=1.5\pm 0.5$, and the whole
galaxy by $n=3\pm 1$. The high-redshift spheroids are compact due to the
dissipative inflow of gas and the high universal density. The stellar surface
density within the effective radius of each galaxy as it evolves remains
roughly constant in time after its first growth. For galaxies of a fixed
stellar mass, the surface density is higher at higher redshifts.
