  The auto-correlation function of the Lyman-$\alpha$ (Ly$\alpha$) forest flux
from high-z quasars can statistically probe all scales of the intergalactic
medium (IGM) just after the epoch of reionization. The thermal state of the
IGM, which is determined by the physics of reionization, sets the amount of
small-scale power seen in the \lya forest. To study the sensitivity of the
auto-correlation function to the thermal state of the IGM, we compute the
auto-correlation function from cosmological hydrodynamical simulations with
semi-numerical models of the thermal state of the IGM. We create mock data sets
of 20 quasars to forecast constraints on $T_0$ and $\gamma$, which characterize
a tight temperature-density relation in the IGM, at $5.4 \leq z \leq 6$. At $z
= 5.4$ we find that an ideal data set constrains $T_0$ to 29\% and $\gamma$ to
9\%. In addition, we investigate four realistic reionization scenarios that
combine temperature and ultra-violet background (UVB) fluctuations at $z =
5.8$. We find that, when using mock data generated from a model that includes
temperature and UVB fluctuations, we can rule out a model with no temperature
or UVB fluctuations at $>1\sigma$ level 50.5\% of the time.

Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies