  Motivated by the significant performance gains which polar codes experience
under successive cancellation list decoding, their scaling exponent is studied
as a function of the list size. In particular, the error probability is fixed
and the trade-off between block length and back-off from capacity is analyzed.
A lower bound is provided on the error probability under $\rm MAP$ decoding
with list size $L$ for any binary-input memoryless output-symmetric channel and
for any class of linear codes such that their minimum distance is unbounded as
the block length grows large. Then, it is shown that under $\rm MAP$ decoding,
although the introduction of a list can significantly improve the involved
constants, the scaling exponent itself, i.e., the speed at which capacity is
approached, stays unaffected for any finite list size. In particular, this
result applies to polar codes, since their minimum distance tends to infinity
as the block length increases. A similar result is proved for genie-aided
successive cancellation decoding when transmission takes place over the binary
erasure channel, namely, the scaling exponent remains constant for any fixed
number of helps from the genie. Note that since genie-aided successive
cancellation decoding might be strictly worse than successive cancellation list
decoding, the problem of establishing the scaling exponent of the latter
remains open.
