  We present results from a mini-survey of relatively high redshift (1.7 < z <
4) gravitationally lensed radio-quiet quasars observed with the Chandra X-ray
Observatory and with XMM-Newton. The lensing magnification effect allows us to
search for changes in quasar spectroscopic and flux variability properties with
redshift over three orders of magnitude in intrinsic X-ray luminosity. It
extends the study of quasar properties to unlensed X-ray flux levels as low as
a few times 10^{-15} erg cm^{-2} s^{-1} in the observed 0.4-8 keV band. We find
a possible correlation between the X-ray powerlaw photon index and X-ray
luminosity of the gravitationally lensed radio-quiet quasar sample. The X-ray
spectral slope steepens as the X-ray luminosity increases. This correlation is
still significant when we combine our data with other samples of radio-quiet
quasars with z > 1.5, especially in the low luminosity range between
10^{43}-10^{45.5} erg s^{-1}. This result is surprising considering that such a
correlation is not found for quasars with redshifts below 1.5. We suggest that
this correlation can be understood in the context of the hot-corona model for
X-ray emission from quasar accretion disks, under the hypothesis that the
quasars in our sample accrete very close to their Eddington limits and the
observed luminosity range is set by the range of black hole masses (this
hypothesis is consistent with recent predictions of semi-analytic models for
quasar evolution). The upper limits of X-ray variability of our relatively high
redshift sample of lensed quasars are consistent with the known correlation
between variability and luminosity observed in Seyfert 1s when this correlation
is extrapolated to the larger luminosities of our sample.
