1. Advice Coins Scott Aaronson

2. PSPACE/coin: Class of problems solvable by a PSPACE machine that can flip an advice coin (heads with probability p, tails with probability 1-p) as many times as it wants Clear that PSPACE/poly PSPACE/coin Other direction? Could PSPACE/coin=ALL?

3. Hellman-Cover 1970: To distinguish a p=1/2 coin from a p=1/2+ coin with constant bias, you need a probabilistic finite automaton with (1/ ) states I.e. you cant detect a less than 1/exp(n) change in p without more than poly(n) bits to record the statisticsregardless of how many times you flip the coin Seems to answer our question! Except that it doesnt

4. First problem: p could be unbelievably small (1/Ackermann(n)), and info could be stored in log(1/p) Second problem: Hellman-Cover theorem is false for quantum finite automata! I can give a QFA with 2 qubits that distinguishes p=1/2 from p=1/2+ for any >0 So question stands: PSPACE/coin=ALL? BQPSPACE/coin=ALL?

5. Main Result: PSPACE/coin, BQPSPACE/coin are both contained in Something/poly Main Idea: Limiting distribution (or quantum state) of an s-state automaton can be expressed in terms of degree-s rational functions of p. These can oscillate at most s times as p goes from 0 to 1. Need to count and compare roots of real polynomials. If everything is doable in NC, then a PSPACE/poly upper bound follows.