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存在具体的下界技术：多项式方法和 adversary 方法。已有大量研究工作致力于使用这些方法给出下界，并将其与基于决策树的其他度量进行比较与关联。本文探究了这些下界在量子查询复杂度上的取值，以及它们与对称函数（可谓布尔函数中最自然、最基本的集合之一）的其他基于决策树的复杂度度量之间的关系。对于任意全对称函数，我们给出了正 adversary 方法对偶以及私密证书博弈复杂度的平方根的显式构造。这表明对于任何对称函数，这两个值不可区分。此外，我们证明了最近引入的谱灵敏度度量对于每个全对称布尔函数均给出与正 adversary 和近似度相同的值。进一步地，我们研究了部分对称函数 Gap Majority 的量子查询复杂度。该函数因有助于理解随机查询复杂度的组合性质而近期备受关注。我们刻画了 Gap Majority 的量子查询复杂度，并利用量子查询复杂度给出了噪声随机查询复杂度（Ben-David 和 Blais, FOCS 2020）的一个下界。最后，我们研究了对称函数中证书复杂度与块灵敏度相对于灵敏度的大小（甚至考虑常数因子）。我们给出了紧的分离结果，即给出了可能分离度的上界，并通过构造函数实现了这些分离。