We show new lower bounds in the \emph{Merlin-Arthur} (MA) communication model and the related \emph{annotated streaming} or stream verification model. The MA communication model is an enhancement of the classical communication model, where in addition to the usual players Alice and Bob, there is an all-powerful but untrusted player Merlin who knows their inputs and tries to convince them about the output. Most functions have MA protocols with total communication significantly smaller than what would be needed without Merlin. We focus on the online MA (OMA) model, which is the MA analogue of one-way communication, and introduce the notion of \emph{non-trivial-OMA} complexity of a function. This is the minimum total communication needed by any non-trivial OMA protocol computing that function, where a trivial OMA protocol is one where Alice sends Bob roughly as many bits as she would have sent without Merlin. We prove a lower bound on the non-trivial-OMA complexity of a natural function \emph{Equals-Index} (basically the well-known Index problem on large domains) and identify it as a canonical problem for proving strong lower bounds on this complexity: reductions from it (i) reproduce and/or improve upon the lower bounds for all functions that were previously known to have large non-trivial-OMA complexity, (ii) exhibit the first explicit functions whose non-trivial-OMA complexity is superlinear, and even exponential, in their classical one-way complexity, and (iii) show functions on input size $n$ for which this complexity is as large as $n/\log n$. While exhibiting a function with $\omega(\sqrt{n})$ (standard) OMA complexity is a longstanding open problem, we did not even know of any function with $\omega(\sqrt{n})$ non-trivial-OMA complexity. We further extend the lower bounds to a related streaming model called annotated streaming.

翻译：我们在Merlin-Arthur（MA）通信模型及相关注释流或流验证模型中证明了新下界。MA通信模型是对经典通信模型的增强，除了通常的参与者Alice和Bob外，存在一位全知但不可信的参与者Merlin，他知晓双方输入并试图说服他们关于输出结果。大多数函数存在总通信量远小于无需Merlin时所需通信量的MA协议。我们聚焦于在线MA（OMA）模型（即MA版本的单向通信），并引入函数的非平凡OMA复杂度概念。该复杂度定义为计算该函数的任何非平凡OMA协议所需的最小总通信量，其中平凡OMA协议是指Alice发送给Bob的比特数大致等于其无需Merlin时发送的比特数。我们证明了自然函数Equals-Index（本质上是大规模域上的经典Index问题）的非平凡OMA复杂度下界，并将其识别为证明该复杂度强下界的典范问题：从该问题的归约（i）复现和/或改进了所有已知具有大非平凡OMA复杂度函数的下界；（ii）首次展示了非平凡OMA复杂度相对于经典单向复杂度呈超线性甚至指数增长的显式函数；（iii）揭示了输入规模为n的函数中该复杂度可达n/log n。尽管证明存在ω(√n)标准OMA复杂度的函数仍是长期未解难题，我们此前甚至不知道任何具有ω(√n)非平凡OMA复杂度的函数。我们进一步将这些下界推广到名为注释流的相关流模型。