  Ar{\i}kan's polar coding technique is based on the idea of synthesizing $n$
channels from the $n$ instances of the physical channel by a simple linear
encoding transformation. Each synthesized channel corresponds to a particular
input to the encoder. For large $n$, the synthesized channels become either
essentially noiseless or almost perfectly noisy, but in total carry as much
information as the original $n$ channels. Capacity can therefore be achieved by
transmitting messages over the essentially noiseless synthesized channels.
Unfortunately, the set of inputs corresponding to reliable synthesized channels
is poorly understood, in particular how the set depends on the underlying
physical channel. In this work, we present two analytic conditions sufficient
to determine if the reliable inputs corresponding to different discrete
memoryless channels are aligned or not, i.e. if one set is contained in the
other. Understanding the alignment of the polarized sets is important as it is
directly related to universality properties of the induced polar codes, which
are essential in particular for network coding problems. We demonstrate the
performance of our conditions on a few examples for wiretap and broadcast
channels. Finally we show that these conditions imply that the simple quantum
polar coding scheme of Renes et al. [Phys. Rev. Lett. 109, 050504 (2012)]
requires entanglement assistance for general channels, but also show such
assistance to be unnecessary in many cases of interest.
