Studies on experimental realization of two-dimensional anyons in terms of quasiparticles have been restricted, so far, to only anyons on the plane. It is known, however, that the geometry and topology of space can have significant effects on quantum statistics for particles moving on it. In this talk, I will show that the emerging fractional statistics for particles restricted to move on the sphere, instead of on the plane, arises naturally in the context of quantum impurity problems. In particular, I will demonstrate a setup in which the lowest-energy spectrum of two linear bosonic molecules immersed in a quantum many particle environment can coincide with the anyonic spectrum on the sphere. This paves the way towards experimental realization of anyons on the sphere using molecular impurities. Finally, I will present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as q-deformed Lie algebras. In particular, I will exemplify this approach by considering a quantum rotor interacting with a bath of bosons.