  We study a family of quantum analogs of L\'evy's stochastic area for planar
Brownian motion depending on a variance parameter $\sigma \geq 1$ which deform
to the classical L\'evy area as $\sigma\rightarrow\infty$. They are defined as
second rank iterated stochastic integrals against the components of planar
Brownian motion, which are one-dimensional Brownian motions satisfying
Heisnberg-type commutation relations. Such iterated integrals can be multiplied
using the sticky shuffle product determined by the underlying It\^o algebra of
stochastic differentials. We use the corresponding Hopf algebra structure to
evaluate the moments of the quantum L\'evy areas and study how they deform to
their classical values, which are well known to be given essentially by the
Euler numbers, in the infinite variance limit.
