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 $zsim7$, which would be a strong indication of a partially neutral IGM.
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