I wonder whether it could be integrated with something like SETI. I'm not so sure PoW is the long-term solution to Crypto anyway, and I am skeptical of SETI's merits, but perhaps signal pattern analysis could be involved somehow.
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SETI is a good candidate for useful but marketless computational work. Generally, anything only the public sector can do would work. Trouble is, there are no universally agreed upon criteria for such work.
I am skeptical of any claims that something is public-sector only. The fact that governments have monopolized a given service is not proof that the service in question can only be provided by the government, or that government is the best solution.