  This paper gives a nearly tight characterization of the quantum communication
complexity of the permutation-invariant Boolean functions. With such a
characterization, we show that the quantum and randomized communication
complexity of the permutation-invariant Boolean functions are quadratically
equivalent (up to a logarithmic factor). Our results extend a recent line of
research regarding query complexity \cite{AA14, Cha19, BCG+20} to communication
complexity, showing symmetry prevents exponential quantum speedups.
  Furthermore, we show the Log-rank Conjecture holds for any non-trivial total
permutation-invariant Boolean function. Moreover, we establish a relationship
between the quantum/classical communication complexity and the approximate rank
of permutation-invariant Boolean functions. This implies the correctness of the
Log-approximate-rank Conjecture for permutation-invariant Boolean functions in
both randomized and quantum settings (up to a logarithmic factor).
