  The Ly$\alpha$ line of high-redshift galaxies has emerged as a powerful probe
of both early galaxy evolution and the epoch of reionization (EoR). Motivated
by the upcoming wide-field survey with the Subaru Hyper Suprime-Cam (HSC), we
study the angular correlation function (ACF) of narrow-band selected,
$z\approx7$ Ly$\alpha$ emitting galaxies (LAEs). The clustering of LAEs is
determined by both: (i) their typical host halo masses, $\bar{M}_{\rm h}$; and
(ii) the absorption due to a patchy EoR, characterized by an average neutral
fraction of the intergalactic medium, $\bar{x}_{\rm HI}$. We bracket the
allowed LAE ACFs by exploring extreme scenarios for both the intrinsic
Ly$\alpha$ emission and the large-scale pattern (i.e. morphology) of cosmic
ionized patches in physical EoR models. Independent of the EoR morphology,
current $z\approx 7$ ACF measurements constrain $\bar{x}_{\rm HI}\lesssim 0.5$
(1-$\sigma$). We also find that the low values of the currently-observed ACF
imply that LAEs are hosted by relatively small dark matter halos: $\bar{M}_{\rm
h} \lesssim 10^{10} M_{\odot}$, with corresponding duty cycles of $\lesssim$
few per cent. These values are over an order of magnitude lower than the
analogous ones for color-selected, Lyman break galaxies, suggesting that
$z\approx 7$ narrow-band LAEs searches are preferentially selecting young,
star-burst galaxies, residing in less massive halos. The upcoming Ultra Deep
campaign with the HSC will significantly improve constraints on both the EoR
and LAE host halos.
