Fault-tolerant consensus is about reaching agreement on some of the input values in a limited time by non-faulty autonomous processes, despite of failures of processes or communication medium. This problem is particularly challenging and costly against an adaptive adversary with full information. Bar-Joseph and Ben-Or (PODC'98) were the first who proved an absolute lower bound $\Omega(\sqrt{n/\log n})$ on expected time complexity of consensus in any classic (i.e., randomized or deterministic) message-passing network with $n$ processes succeeding with probability $1$ against such a strong adaptive adversary crashing processes. Seminal work of Ben-Or and Hassidim (STOC'05) broke the $\Omega(\sqrt{n/\log n})$ barrier for consensus in classic (deterministic and randomized) networks by employing quantum computing. They showed an (expected) constant-time quantum algorithm for a linear number of crashes $t<n/3$. In this paper, we improve upon that seminal work by reducing the number of quantum and communication bits to an arbitrarily small polynomial, and even more, to a polylogarithmic number -- though, the latter in the cost of a slightly larger polylogarithmic time (still exponentially smaller than the time lower bound $\Omega(\sqrt{n/\log n})$ for classic computation).

翻译：容错共识是指在存在进程或通信介质故障的情况下，无故障自主进程在有限时间内对其部分输入值达成一致。这一问题在面对具备完全信息的自适应强敌手时尤为困难和耗时。Bar-Joseph与Ben-Or（PODC'98）率先证明：在任何含n个进程且以概率1成功对抗此类强自适应崩溃敌手的经典（即随机化或确定性）消息传递网络中，共识的期望时间复杂度存在绝对下界$\Omega(\sqrt{n/\log n})$。Ben-Or与Hassidim（STOC'05）的开创性工作通过引入量子计算，突破了经典（确定性和随机化）网络中共识问题的$\Omega(\sqrt{n/\log n})$壁垒。该研究针对线性数量崩溃（$t<n/3$）场景给出了（期望）常数时间的量子算法。本文改进了这一开创性成果：将量子比特与通信比特数降至任意小的多项式量级，甚至进一步压缩至多对数数量级——尽管后者需要以略大的多对数时间开销为代价（该时间仍远小于经典计算的下界$\Omega(\sqrt{n/\log n})$）。