  X-ray emission from QSOs has been used to assess SMBH accretion properties up
to $z$~6. However, at $z>6$ only ~15 QSOs are covered by sensitive X-ray
observations, preventing a statistically significant investigation of the X-ray
properties of QSOs in the first Gyr of the Universe. We present new Chandra
observations of 10 $z>6$ QSOs, selected to have virial black-hole mass
estimates from Mg II line spectroscopy. Adding archival X-ray data for an
additional 15 $z>6$ QSOs, we investigate the X-ray properties of the QSO
population in the first Gyr of the Universe, focusing in particular on the
$L_{UV}-L_{X}$ relation, which is traced by the $\alpha_{ox}$ parameter, and
the shape of their X-ray spectra. We performed photometric analyses to derive
estimates of the X-ray luminosities, and thus the $\alpha_{ox}$ values and
bolometric corrections ($K_{bol}=L_{bol}/L_{X}$). We compared the resulting
$\alpha_{ox}$ and $K_{bol}$ distributions with the results found for QSO
samples at lower redshift. Finally, we performed a basic X-ray spectral
analysis of the brightest $z>6$ QSOs to derive their individual photon indices,
and joint spectral analysis of the whole sample to estimate the average photon
index. We confirm a lack of significant evolution of $\alpha_{ox}$ with
redshift, extending the results from previous works up to $z>6$, and the trend
of an increasing bolometric correction with increasing luminosity found for
QSOs at lower redshifts. The average power-law photon index of our sample
($\Gamma=2.20_{-0.34}^{+0.39}$ and $\Gamma=2.13_{-0.13}^{+0.13}$ for sources
with $<30$ and $>30$ net counts, respectively) is slightly steeper than, but
still consistent with, typical QSOs at $z=1-6$. All these results point toward
a lack of substantial evolution of the inner accretion-disk/hot-corona
structure in QSOs from low redshift to $z>6$. Our data hint at generally high
Eddington ratios at $z>6$.
