Symmetry principles are fundamental to many areas of physics, and quantum information
science is no exception to this rule. Geometry, and in particular convex geometry,
is a central tool in quantum information theory which has recently proven invaluable
in the context of characterising distinctly quantum resources. At a highest level this
thesis seeks to contribute to the following question: How do symmetry principles and
geometry constrain the processing of quantum information?
The research presented in this thesis splits into three main parts. To start with we
focus on finding finite conditions for symmetry constrained quantum dynamics. In this
context, we derive finite and tractable sufficient conditions for the resource theory of
asymmetry, which we then apply to problems in covariant quantum error correction
and thermal processing. We then turn our attention to exploring geometric features of
quantum states in thermodynamic equilibrium and beyond. In particular, we present
a 2d planar geometric representation of quasiclassical states of a system with a fixed
Hamiltonian, where the area of this representation measures the deviation from the
manifold of Gibbs states. This simple geometric picture allows us to recast a known proof
of the Gibbs state in a clean and intuitive way. Finally, we focus our attention on magic
states, which constitute a key building block in most leading schemes for fault-tolerant
quantum computing. We further develop recent majorization toolsets for working with
quasiprobability representations of magic states, which we subsequently use to derive
fundamental trade-off relations on distillation protocols based on stabilizer codes.
Notably, these results apply to qubit magic distillation protocols under the restriction
to CSS codes.