  We present and analyse the deepest, high-quality Ks-band images ever obtained
of luminous quasars at z~4, in an attempt to determine the basic properties of
their host galaxies less than 1 Gyr after the first recorded appearance of
black holes with Mbh > 10^9 Msol. To maximise the robustness of our results we
have carefully selected two SDSS quasars at z~4. These quasars are
representative of the most luminous quasars known at this epoch but they also,
crucially, lie within 40 arcsec of comparably-bright foreground stars (required
for accurate PSF definition), and have redshifts which ensure line-free Ks-band
imaging. The data were obtained in excellent seeing (<0.4-arcsec) at the ESO
VLT with integration times of ~5.5 hours per source. Via carefully-controlled
separation of host-galaxy and nuclear light, we estimate the luminosities and
stellar masses of the host galaxies, and set constraints on their half-light
radii. The quasar host galaxies have K-band luminosities similar to radio
galaxies at comparable redshifts, suggesting that these quasar hosts are also
among the most massive galaxies in existence at this epoch. However, the quasar
hosts are a factor ~5 smaller than the host galaxies of luminous low-redshift
quasars. We estimate the stellar masses of the z~4 host galaxies to lie in the
range 2-10x10^11 Msol, and use the CIV emission line in the Sloan spectra to
estimate the masses of their black holes. The results imply a
black-hole:host-galaxy mass ratio Mbh:Mgal~0.01-0.05. This is an order of
magnitude higher than typically seen in the low-redshift Universe, and is
consistent with existing evidence for a systematic growth in this mass ratio
with increasing redshift, at least for objects selected as powerful AGN.
