  Constraining the escape fraction of Lyman Continuum (LyC) photons from
high-redshift galaxies is crucial to understanding reionization. Recent
observations have demonstrated that various characteristics of the Ly$\alpha$
emission line correlate with the inferred LyC escape fraction ($f_{\rm
esc}^{\rm LyC}$) of low-redshift galaxies. Using a data-set of 9,600 mock
Ly$\alpha$ spectra of star-forming galaxies at $4.64 \leq z \leq 6$ from the
SPHINX$^{20}$ cosmological radiation hydrodynamical simulation, we study the
escape of Ly$\alpha$ and LyC photons. We find that our mock Ly$\alpha$
observations are representative of high-redshift observations and that typical
observational methods tend to over-predict the Ly$\alpha$ escape fraction
($f_{\rm esc}^{\rm Ly\alpha}$) by as much as two dex. We investigate the
correlations between $f_{\rm esc}^{\rm LyC}$ and $f_{\rm esc}^{\rm Ly\alpha}$,
Ly$\alpha$ equivalent width ($W_{\lambda}({\rm Ly\alpha})$), peak separation
($v_{\rm sep}$), central escape fraction ($f_{\rm cen}$), and red peak
asymmetry ($A_f^{\rm red}$). We find that $f_{\rm esc}^{\rm Ly\alpha}$ and
$f_{\rm cen}$ are good diagnostics for LyC leakage, selecting for galaxies with
lower neutral gas densities and less UV attenuation that have recently
experienced supernova feedback. In contrast, $W_{\lambda}({\rm Ly\alpha})$ and
$v_{\rm sep}$ are found to be necessary but insufficient diagnostics, while
$A_f^{\rm red}$ carries little information. Finally, we use stacks of
Ly$\alpha$, H$\alpha$, and F150W mock surface brightness profiles to find that
galaxies with high $f_{\rm esc}^{\rm LyC}$ have less extended Ly$\alpha$ and
F150W but larger H$\alpha$ haloes than their non-leaking counterparts. This
confirms that Ly$\alpha$ spectral profiles and surface brightness morphology
can be used to better understand the escape of LyC photons from galaxies during
the Epoch of Reionization.
