  This paper explores a fine-grained version of the Watrous conjecture,
including the randomized and quantum algorithms with success probabilities
arbitrarily close to $1/2$. Our contributions include the following:
  i) An analysis of the optimal success probability of quantum and randomized
query algorithms of two fundamental partial symmetric Boolean functions given a
fixed number of queries. We prove that for any quantum algorithm computing
these two functions using $T$ queries, there exist randomized algorithms using
$\mathsf{poly}(T)$ queries that achieve the same success probability as the
quantum algorithm, even if the success probability is arbitrarily close to 1/2.
  ii) We establish that for any total symmetric Boolean function $f$, if a
quantum algorithm uses $T$ queries to compute $f$ with success probability
$1/2+\beta$, then there exists a randomized algorithm using $O(T^2)$ queries to
compute $f$ with success probability $1/2+\Omega(\delta\beta^2)$ on a
$1-\delta$ fraction of inputs, where $\beta,\delta$ can be arbitrarily small
positive values. As a corollary, we prove a randomized version of
Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the
regime where the success probability of algorithms can be arbitrarily close to
1/2.
  iii) We present polynomial equivalences for several fundamental complexity
measures of partial symmetric Boolean functions. Specifically, we first prove
that for certain partial symmetric Boolean functions, quantum query complexity
is at most quadratic in approximate degree for any error arbitrarily close to
1/2. Next, we show exact quantum query complexity is at most quadratic in
degree. Additionally, we give the tight bounds of several complexity measures,
indicating their polynomial equivalence.
