Context. X-ray emission from quasars (QSOs) has been used to assess supermassive black hole (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 the QSO population in the first Gyr of the Universe. Aims. We present new Chandra observations of a sample of 10 z > 6 QSOs, selected to have virial black-hole mass estimates from Mg II line spectroscopy (log M BH M = 8.5 − 9.6). 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 α ox parameter, and the shape of their X-ray spectra. Methods. We performed photometric analyses to derive estimates of the X-ray luminosities of our z > 6 QSOs, and thus their α ox values and bolometric corrections (K bol = L bol /L X ). We compared the resulting α ox and K bol distributions with the results found for QSO samples at lower redshift, and ran several statistical tests to check for a possible evolution of the L UV − L X relation. 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. Results. We detected seven of the new Chandra targets in at least one standard energy band, while two more are detected discarding energies E > 5 keV, where background dominates. We confirm a lack of significant evolution of α ox with redshift, extending the results from previous works up to z > 6 with a statistically significant QSO sample, 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 ( Γ = 2.20 +0.39 −0.34 and Γ = 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. Conclusions. 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.