The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.

翻译：查询模型在复杂性理论和量子计算中占据重要地位的主要原因在于其具有具体的下界技术：多项式法和对手法。学界一直致力于运用这些方法给出下界，并将其与基于决策树的其他测度进行比较与关联。我们探究了这些下界在量子查询复杂度中的数值，以及它们与对称函数类（可论证地，这是最自然且最基本的布尔函数集合之一）中其他基于决策树的复杂度测度之间的关系。我们给出了任意全对称函数的正对手法对偶以及私有硬币证书博弈复杂度的平方根的显式构造，表明这两个数值在对称函数上无法区分。此外，我们发现新近引入的谱灵敏度测度与每个全对称布尔函数的正对手度和近似阶均取值相同。进一步地，我们研究了部分对称函数——间隙多数函数的量子查询复杂度。该函数近期在理解随机化查询复杂度的复合特性方面获得了重要关注。我们刻画了间隙多数函数的量子查询复杂度，并给出了其噪声随机化查询复杂度（Ben-David 和 Blais, FOCS 2020）关于量子查询复杂度的下界。最后，我们研究了对称函数中证书复杂度与区块灵敏度相较于灵敏度所能达到的比值（甚至包括常数因子差异），给出了紧致分离结果（即分离程度的上界），并构造了达到该上界的函数。