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

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... ties 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. We performed photometric analyses to derive estimates of the X-ray luminosities, and thus the 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. 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 α_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 (Γ=2.20_-0.34^+0.39 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. 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.