  In a recent breakthrough result, Chattopadhyay, Mande and Sherif [ECCC
TR18-17] showed an exponential separation between the log approximate rank and
randomized communication complexity of a total function $f$, hence refuting the
log approximate rank conjecture of Lee and Shraibman [2009]. We provide an
alternate proof of their randomized communication complexity lower bound using
the information complexity approach. Using the intuition developed there, we
derive a polynomially-related quantum communication complexity lower bound
using the quantum information complexity approach, thus providing an
exponential separation between the log approximate rank and quantum
communication complexity of $f$. Previously, the best known separation between
these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain,
Kothari and Lee [CCC, 2017]. This settles one of the main question left open by
Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank
conjecture of Lee and Shraibman [2009]. Along the way, we develop a
Shearer-type protocol embedding for product input distributions that might be
of independent interest.
