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Suppose there is a material that interacts with 50% of high-energy neutrinos. To what temperature such material will heat up naturally here on Earth? What power can be extracted from it?

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The obvious use of this material would be putting it around reactor cores, I presume. en.wikipedia.org/wiki/Neutrino#Artificial - 185 MW of neutrino energy loss out of 4000 MW thermal output (in a 1.3GW plant). –  Deer Hunter Sep 7 '13 at 7:46
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@DeerHunter Well, reactor neutrinos are not "high energy" in the usual terminology. They are typically a few MeV. –  dmckee Sep 7 '13 at 11:48
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I'm concerned that the question violates the "fictional physics" constraint. Above the Z mass, neutrinos do interact strongly matter, but that is uninteresting for the purposes of this question because the main neutrino flux through the Earth is at solar energies (which, like reactor energies are a few MeV). The cross-section there is low for very fundamental reasons relating to the mass of the weak bosons and the Fermi coupling constant. –  dmckee Sep 7 '13 at 11:51
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Have a look at the neutrino slides of this talk: users.ictp.it/~smr2246/monday/gaisser-NUSKY.pdf . The numbers are very small there is no energy pool in the high energy neutrino flux and in addition, due to the weak interaction the by products take away most of the energy, except for electron neutrino interactions. –  anna v Sep 7 '13 at 13:21

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