  We study the prospects for constraining the ionized fraction of the
intergalactic medium (IGM) at $z>6$ with the next generation of large
Ly$\alpha$ emitter surveys. We make predictions for the upcoming Subaru Hyper
Suprime-Cam (HSC) Ly$\alpha$ survey and a hypothetical spectroscopic survey
performed with the James Webb Space Telescope (JWST). Considering various
scenarios where the observed evolution of the Ly$\alpha$ luminosity function of
Ly$\alpha$ emitters at $z>6$ is explained partly by an increasingly neutral IGM
and partly by intrinsic galaxy evolution, we show how clustering measurements
will be able to distinguish between these scenarios. We find that the HSC
survey should be able to detect the additional clustering induced by a neutral
IGM if the global IGM neutral fraction is greater than $\sim$20 per cent at
$z=6.5$. If measurements of the Ly$\alpha$ equivalent widths (EWs) are also
available, neutral fractions as small as 10 per cent may be detectable by
looking for correlation between the EW and the local number density of objects.
In this case, if it should turn out that the IGM is significantly neutral at
$z=6.5$ and the intrinsic EW distribution is relatively narrow, the observed
EWs can also be used to construct a map of the locations and approximate sizes
of the largest ionized regions. For the JWST survey, the results appear a bit
less optimistic. Since such surveys probe a large range of redshifts, the
effects of the IGM will be mixed up with any intrinsic galaxy evolution that is
present, making it difficult to disentangle the effects. However, we show that
a survey with the JWST will have a possibility of observing a large group of
galaxies at $z\sim7$, which would be a strong indication of a partially neutral
IGM.
