  We revisit the classical online portfolio selection problem. It is widely
assumed that a trade-off between computational complexity and regret is
unavoidable, with Cover's Universal Portfolios algorithm, SOFT-BAYES and
ADA-BARRONS currently constituting its state-of-the-art Pareto frontier. In
this paper, we present the first efficient algorithm, BISONS, that obtains
polylogarithmic regret with memory and per-step running time requirements that
are polynomial in the dimension, displacing ADA-BARRONS from the Pareto
frontier. Additionally, we resolve a COLT 2020 open problem by showing that a
certain Follow-The-Regularized-Leader algorithm with log-barrier regularization
suffers an exponentially larger dependence on the dimension than previously
conjectured. Thus, we rule out this algorithm as a candidate for the Pareto
frontier. We also extend our algorithm and analysis to a more general problem
than online portfolio selection, viz. online learning of quantum states with
log loss. This algorithm, called SCHRODINGER'S BISONS, is the first efficient
algorithm with polylogarithmic regret for this more general problem.
