Find me a document for which this predicate is true (...).
More practically, find me a document signed by this public key (...), and whose content begins with this prefix (...). This makes it possible to retrieve key-values.
Added new twist: here is a promise of a prize to whomever first provides the document. Inspired by how Bitcoin transactions are secured by two halves of a puzzle.
These are generic mechanisms for a truly distributed data storage ecosystem. Data gets inserted into the cloud, and storers get paid for correctly responding to queries.
Deliberately left unstated is how insertion happens, how storers choose what to store, how the query happens, etc. Market mechanisms will figure this out.
For the use case of private, encrypted data inserted into the public cloud, there does occur the interesting economic monopsony problem that there will only be one consumer for a given piece of data, the holder of the decryption key.
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