## Thursday, May 09, 2019

### Multiple Provers

Just over thirty years ago on May 5, 1989, I defended my PhD Thesis Complexity-Theoretic Aspects of Interactive Proof Systems. It's starts off with a parable for interactive proof systems.
Victor, a venture capitalist, had everything a man could desire: money, women and power. But he felt something missing. He decided he lacked knowledge. So Victor packed up his bags and headed to the Himalayas in search of ultimate truths. The natives pointed Victor to a tall mountain and mentioned rumors of a great man full of wisdom. Victor, who smartly brought some climbing equipment, tackled the mountain until he reached a small cave near the summit. Victor found the great Pulu, grand guru of all that is known. Victor inquired to some ultimate truths and Pulu responded, I will teach you but you must not trust my words. Victor agreed and found he learned much even though he had to verify all the sayings of the great Pulu. Victor though lacked complete happiness and he asked if he could learn knowledge beyond what he could learn in this manner. The grand guru replied, You may ask and I will answer. Victor pondered this idea for a minute and said, "Since you know all that is known, why can you not predict my questions?" A silence reigned over the mountain for a short while until the guru finally spoke, You must use other implements, symbols of your past life. Victor thought for a while and reached into his backpack and brought out some spare change he had unwittingly carried with him. Even the great Pulu can not predict the flip of a coin. He started flipping the coins to ask the guru and wondered what can I learn now?
Without the coins, one gets the complexity class NP. My thesis didn't answer the last question, but by the end of the year, Shamir building on work of Lund, Fortnow, Karloff and Nisan showed this class IP was equal to PSPACE, the problems we could solve in a polynomial amount of memory.

Part of my thesis explored the class MIP where we had multiple Pulus (provers) on different mountain tops unable to communicate. The news was disappointing, we failed to get a PSPACE upper bound for MIP, only NEXP (nondeterministic exponential time) and our proof that two provers sufficed relied on a bad assumption on how errors get reduced when you run multiple protocols in parallel. Later Babai, Lund and myself showed MIP = NEXP and Ran Raz showed parallel repetition does reduce the error sufficiently.

Back in the 80's we didn't even imagine the possibility that the Pulus had shared entangled quantum bits. Does the entanglement allow the provers to cheat or can the entanglement allow them to prove more things? Turns out to be much more, as a new result by Anand Natarajan and John Wright shows that MIP*, MIP with classical communication, classical verifier and two provers with previously entangled quantum bits, can compute everything in NEEXP, nondeterministic double exponential time. This is only a lower bound for MIP*, possibly one can do even more.

Neat to see my three-decade old thesis explored ideas that people are still thinking about today.