We study the hard-X-ray spectral properties of ten highly luminous radio-quiet (RQ) active galactic nuclei (AGNs) at z=1.3-3.2, including new XMM-Newton observations of four of these sources. We find a significant correlation between the normalized accretion rate (L/L_Edd) and the hard-X-ray photon index (Gamma) for 35 moderate-high luminosity RQ AGNs including our ten highly luminous sources. Within the limits of our sample, we show that a measurement of Gamma and L_X can provide an estimate of L/L_Edd and black-hole (BH) mass (M_BH) with a mean uncertainty of a factor of <~3 on the predicted values of these properties. This may provide a useful probe for tracing the history of BH growth in the Universe, utilizing samples of X-ray-selected AGNs for which L/L_Edd and M_BH have not yet been determined systematically. It may prove to be a useful way to probe BH growth in distant Compton-thin type 2 AGNs. We also find that the optical-X-ray spectral slope (a_ox) depends primarily on optical-UV luminosity rather than on L/L_Edd in a sample of RQ AGNs spanning five orders of magnitude in luminosity and over two orders of magnitude in L/L_Edd. We detect a significant Compton-reflection continuum in two of our highly luminous sources, and in the stacked X-ray spectrum of seven other sources with similar luminosities, we obtain a mean relative Compton reflection of R=0.9^{+0.6}_{-0.5} and an upper limit on the rest-frame equivalent width of a neutral Fe Ka line of 105 eV. We do not detect a significant steepening of the X-ray power-law spectrum below rest-frame 2 keV in any of our highly luminous sources, suggesting that a soft-excess feature, commonly observed in local AGNs, either does not depend strongly on L/L_Edd, or is not accessible at high redshifts using current X-ray detectors. [Abridged]
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