  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.
