Quantum computers can execute algorithms that dramatically outperform classical
computation. As the best-known example, Shor discovered an efficient quantum algorithm …

This article surveys quantum computational complexity, with a focus on three fundamental
notions: polynomial-time quantum computations, the efficient verification of quantum proofs …

In the first part of the paper we introduce a new theorem-proving procedure, that is a new
efficient method of communicating a proof. Any such method implies, directly or indirectly, a …

Public key cryptography is a major interdisciplinary subject with many real-world
applications, such as digital signatures. A strong background in the mathematics underlying …

This paper considers the computational complexity of the discrete logarithm and related
problems in the context of “generic algorithms”—that is, algorithms which do not exploit any …

In a previous paper [BS] we proved, using the elements of the theory of nilpotent groups, that
some of the fundamental computational problems in matriz groups belong to NP. These …

The origins of computation group theory (CGT) date back to the late 19th and early 20th
centuries. Since then, the field has flourished, particularly during the past 30 to 40 years, and …

ARTHUR-MERLINGAMES 255 result, one obtains the strongest evidence yet against NP-
completeness of graph isomorphism and coset intersection. Using random oracles, we …

These are lecture notes from a weeklong course in quantum complexity theory taught at the
Bellairs Research Institute in Barbados, February 21-25, 2016. The focus is quantum circuit …

From the winner of the Turing Award and the Abel Prize, an introduction to computational
complexity theory, its connections and interactions with mathematics, and its central role in …