  Polar codes are the first capacity achieving and efficiently implementable
codes for classical communication. Recently they have also been generalized to
communication over classical-quantum and quantum channels. In this work we
present our recent results for polar coding in quantum information theory,
including applications to classical-quantum multiple access channels,
interference channels and compound communication settings, including the first
proof of channel coding achieving the Han-Kobayashi rate region of the
interference channel without the need of a simultaneous decoder. Moreover we
add to the existing framework by extending polar codes to achieve the
asymmetric capacity and improving the block error probability for
classical-quantum channels. In addition we use polar codes to prove a new
achievable rate region for the classical-quantum broadcast channel. We also
discuss polar codes for quantum communication over quantum channels and state
results towards codes for compound quantum channels in this setting. We
conclude by stating a list of interesting open questions to invite further
research on the topic.
