Shannon quantum information entropies Sρ,γ, Fisher informations Iρ,γ, Onicescu energies Oρ,γ and complexities eSO are calculated both in the position (subscript ρ) and momentum (γ) spaces for the azimuthally symmetric two-dimensional nanoring that is placed into the combination of the transverse uniform magnetic field B and the Aharonov–Bohm (AB) flux ϕAB and whose potential profile is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. The increasing intensity B flattens momentum waveforms Φnm(k) and in the limit of the infinitely large fields they turn to zero: Φnm(k)→0 at B→∞, what means that the position wave functions Ψnm(r), which are their Fourier counterparts, tend in this limit to the δ-functions. Position (momentum) Shannon entropy depends on the field B as a negative (positive) logarithm of ωeff≡(ω02+ωc2/4)1/2, where ω0 determines the quadratic steepness of the confining potential and ωc is a cyclotron frequency. This makes the sum Sρnm+Sγnm a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and does satisfy entropic uncertainty relation. Position Fisher information does not depend on m, linearly increases with n and varies as ωeff whereas its n and m dependent Onicescu counterpart Oρnm changes as ωeff−1. The products IρnmIγnm and OρnmOγnm are B-independent quantities. A dependence of the measures on the ring geometry is discussed. It is argued that a variation of the position Shannon entropy or Onicescu energy with the AB field uniquely determines an associated persistent current as a function of ϕAB at B=0. An inverse statement is correct too.