  The discovery of quasars with redshifts higher than six has prompted a great
deal of discussion in the literature regarding the role of quasars, both as
sources of reionization, and as probes of the ionization state of the IGM.
However the extreme ultra-violet (EUV) spectral index cannot be measured
directly for high redshift quasars owing to absorption at frequencies above the
Lyman limit, and as a result, studies of the impact of quasars on the
intergalactic medium during reionization must assume a spectral energy
distribution in the extreme ultra-violet based on observations at low redshift,
z<1. In this paper we use regions of high Ly-alpha transmission (near-zones)
around the highest redshift quasars to measure the quasar EUV spectral index at
z~6. We jointly fit the available observations for variation of near-zone size
with both redshift and luminosity, and propose that the observed relation
provides evidence for an EUV spectral index that varies with absolute magnitude
in the high redshift quasar sample, becoming softer at higher luminosity. Using
a large suite of detailed numerical simulations we find that the typical value
of spectral index for a luminous quasar at z~6 is constrained to be
alpha=1.3+/-0.4 for a specific luminosity of the form L\propto\nu^{-alpha}. We
find the scatter in spectral index among individual quasars to be in the range
~0.75-1.25. These values are in agreement with direct observations at low
redshift, and indicate that there has been no significant evolution in the EUV
spectral index of quasars over 90% of cosmic time.
