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Convince means the verifier can prove to itself what the provers are telling it in polynomial time, without trusting the provers.

It's still unclear whether the algorithm the verifier uses has actually been identified or has only been proven to exist.

>> Still interesting even if it take a computer the size of the earth.

There are some white mice on Magrathea who hold the IP on this.

That sounds like a probabilistically verifiable proof. See http://people.csail.mit.edu/madhu/papers/2009/pcpcacm.pdf for more.