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 \(z>6\).