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.
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