  This PhD thesis is dedicated to the study of the interplay between symmetries
of quantum states and their self-similar properties. It consists of three
connected threads of research: polynomial invariants for multiphoton states,
visualization schemes for quantum many-body systems and a complex networks
approach to quantum walks on a graph.
  First, we study the problem of which many-photon states are equivalent up to
the action of passive linear optics. We prove that it can be converted into the
problem of equivalence of two permutation-symmetric states, not necessarily
restricted to the same operation on all parties. We show that the problem can
be formulated in terms of symmetries of complex polynomials of many variables,
and provide two families of invariants, which are straightforward to compute
and provide analytical results.
  Second, we study a family of recursive visualization schemes for
many-particle systems, for which we have coined the name 'qubism'. While all
many-qudit states can be plotted with qubism, it is especially useful for spin
chains and one-dimensional translationally invariant states. This symmetry
results in self-similarity of the plot, making it more comprehensible and
allowing to discover certain structures.
  Third, we study quantum walks of a single particle on graphs, which are
classical analogues of random walks. Our focus is on the long-time limit of the
probability distribution and we study how (especially in the long-time limit)
off-diagonal elements of the density matrix behave. We use them to perform
quantum community detection - splitting of a graph into subgraphs in such a way
that the coherence between them is small. Our method captures properties that
classical methods cannot - the impact of constructive and destructive
interference, as well as the dependence of the results on the tunneling phase.
