  Context: Strings and other alternative theories describing the quantum
properties of space-time suggest that space-time could present a foamy
structure and also that, in certain cases, quantum gravity (QG) may manifest at
energies much below the Planck scale. One of the observable effects could be
the degradation of the diffraction images of distant sources.
  Aims: We searched for this degradation effect, caused by QG fluctuations, in
the light of the farthest quasars (QSOs) observed by the Hubble Space Telescope
with the aim of setting new limits on the fluctuations of the space-time foam
and QG models.
  Methods: We developed a software that estimates and compares the phase
variation in the interference patterns of the high-redshift QSOs, taken from
the snapshot survey of HST-SDSS, with those of stars that are expected to not
be affected by QG effects. We used a two-parameter function to determine, for
each test star and QSO, the maximum of the diffraction pattern and to calculate
the Strehl ratio.
  Results: Our results go far beyond those already present in the literature.
By adopting the most conservative approach where the correction terms, that
describe the possibility for space-time fluctuations cumulating across long
distances and partially compensate for the effects of the phase variations, are
taken into account. We exclude the random walk model and most of the
holographic models of the space-time foam. Without considering these correction
terms, all the main QG scenarios are excluded. Finally, our results show the
absence of any directional dependence of QG effects and the validity of the
cosmological principle with an independent method; that is, viewed on a large
scale, the properties of the Universe are the same for all observers, including
the effects of space-time fluctuations.
