On the Impossibility of Probabilistic Proofs in Relativized Worlds

We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle. This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it. We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent

Quantum Weakly Nondeterministic Communication Complexity

We study the weakest model of quantum nondeterminism in which a classical proof has to be checked with probability one by a quantum protocol. We show the first separation between classical nondeterministic communication complexity and this model of quantum nondeterministic communication complexity for a total function. This separation is quadratic.Comment: 12 pages. v3: minor correction

Exponential Separation of Quantum and Classical Online Space Complexity

Although quantum algorithms realizing an exponential time speed-up over the best known classical algorithms exist, no quantum algorithm is known performing computation using less space resources than classical algorithms. In this paper, we study, for the first time explicitly, space-bounded quantum algorithms for computational problems where the input is given not as a whole, but bit by bit. We show that there exist such problems that a quantum computer can solve using exponentially less work space than a classical computer. More precisely, we introduce a very natural and simple model of a space-bounded quantum online machine and prove an exponential separation of classical and quantum online space complexity, in the bounded-error setting and for a total language. The language we consider is inspired by a communication problem (the set intersection function) that Buhrman, Cleve and Wigderson used to show an almost quadratic separation of quantum and classical bounded-error communication complexity. We prove that, in the framework of online space complexity, the separation becomes exponential.Comment: 13 pages. v3: minor change

A Casual Tour Around a Circuit Complexity Bound

I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News, September 201