  Recently, the Lyman-$\alpha$ (Ly$\alpha$) forest flux auto-correlation
function has been shown to be sensitive to the mean free path of
hydrogen-ionizing photons, $\lambda_{\text{mfp}}$, for simulations at $z \geq
5.4$. Measuring $\lambda_{\text{mfp}}$ at these redshifts will give vital
information on the ending of reionization. Here we present the first
observational measurements of the Ly$\alpha$ forest flux auto-correlation
functions in ten redshift bins from $5.1 \leq z \leq 6.0$. We use a sample of
35 quasar sightlines at $z > 5.7$ from the extended XQR-30 data set, this data
has signal-to-noise ratios of $> 20$ per spectral pixel. We carefully account
for systematic errors in continuum reconstruction, instrumentation, and
contamination by damped Ly$\alpha$ systems. With these measurements, we
introduce software tools to generate auto-correlation function measurements
from any simulation. For an initial comparison, we show our auto-correlation
measurements with simulation models for recently measured
$\lambda_{\text{mfp}}$ values and find good agreements. Further work in
modeling and understanding the covariance matrices of the data is necessary to
get robust measurements of $\lambda_{\text{mfp}}$ from this data.
