  The temperature of the low-density intergalactic medium (IGM) at high
redshift is sensitive to the timing and nature of hydrogen and HeII
reionization, and can be measured from Lyman-alpha forest absorption spectra.
Since the memory of intergalactic gas to heating during reionization gradually
fades, measurements as close as possible to reionization are desirable. In
addition, measuring the IGM temperature at sufficiently high redshifts should
help to isolate the effects of hydrogen reionization since HeII reionization
starts later, at lower redshift. Motivated by this, we model the IGM
temperature at z>5 using semi-numeric models of patchy reionization. We
construct mock Lyman-alpha forest spectra from these models and consider their
observable implications. We find that the small-scale structure in the
Lyman-alpha forest is sensitive to the temperature of the IGM even at redshifts
where the average absorption in the forest is as high as 90%. We forecast the
accuracy at which the z~5 IGM temperature can be measured using existing
samples of high resolution quasar spectra, and find that interesting
constraints are possible. For example, an early reionization model in which
reionization ends at z~10 should be distinguishable -- at high statistical
significance -- from a lower redshift model where reionization completes at
z~6. We discuss improvements to our modeling that may be required to robustly
interpret future measurements.
