I don't know.

I was thinking something like write a bunch of random numbers to disk, and prove that you have the ability to retrieve any of them at will.

This is obviously not a complete solution. I was asking if anyone knew of any such system.

A positive, not negative, feature of the system is you must be able to rewrite all the storage each time. So the faster your can write the more blocks you win, but also the more storage you have the more blocks you win (and it's a tradeoff between them).