  We explore the interplay between symmetry and randomness in quantum
information. Adopting a geometric approach, we consider states as
$H$-equivalent if related by a symmetry transformation characterized by the
group $H$. We then introduce the Haar measure on the homogeneous space
$\mathbb{U}/H$, characterizing true randomness for $H$-equivalent systems.
While this mathematical machinery is well-studied by mathematicians, it has
seen limited application in quantum information: we believe our work to be the
first instance of utilizing homogeneous spaces to characterize symmetry in
quantum information. This is followed by a discussion of approximations of true
randomness, commencing with $t$-wise independent approximations and defining
$t$-designs on $\mathbb{U}/H$ and $H$-equivalent states. Transitioning further,
we explore pseudorandomness, defining pseudorandom unitaries and states within
homogeneous spaces. Finally, as a practical demonstration of our findings, we
study the expressibility of quantum machine learning ansatze in homogeneous
spaces. Our work provides a fresh perspective on the relationship between
randomness and symmetry in the quantum world.
