  Observations of high-redshift quasars provide information on the massive
black holes (MBHs) powering them and the galaxies hosting them. Current
observations of $z \gtrsim 6$ hosts, at sub-mm wavelengths, trace the
properties of cold gas, and these are used to compare with the correlations
between MBHs and galaxies characterising the $z=0$ population. The relations at
$z=0$, however, rely on stellar-based tracers of the galaxy properties. We
perform a very-high resolution cosmological zoom-in simulation of a $z=7$
quasar including state-of-the-art non-equilibrium chemistry, MBH formation,
growth and feedback, to assess the evolution of the galaxy host and the central
MBH, and compare the results with recent ALMA observations of high-redshift
quasars. We measure both the stellar-based quantities used to establish the
$z=0$ correlations, as well as the gas-based quantities available in $z \gtrsim
6$ observations, adopting the same assumptions and techniques used in
observational studies. The high-redshift studies argued that MBHs at high
redshift deviate from the local MBH-galaxy correlations. In our analysis of the
single galaxy we evolve, we find that the high-redshift population sits on the
same correlations as the local one, when using the same tracers used at $z=0$.
When using the gas-based tracers, however, MBHs appear to be over-massive. The
discrepancy between local and high-redshift MBHs seems caused by the different
tracers employed, and necessary assumptions, and not by an intrinsic
difference. Better calibration of the tracers, higher resolution data and
availability of facilities that can probe the stellar population will be
crucial to assess precisely and accurately high-redshift quasar hosts.
