Context : X-ray emission from quasars ( QSOs ) has been used to assess supermassive black hole ( SMBH ) accretion properties up to z \approx 6 . However , at z > 6 only \approx 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 \frac { M _ { \mathrm { BH } } } { M _ { \odot } } = 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 \alpha _ { 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 \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 , 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 \alpha _ { 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 ( \langle \Gamma \rangle = 2.20 _ { -0.34 } ^ { +0.39 } and \langle \Gamma \rangle = 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 .