Abstract <p>This work uses an efficient finite Whittaker Cardinal function representation-based approximation scheme to expose hidden bound states and Shannon information entropies of Schrödinger equation with Kepler–Coulomb potential and its rational extensions in curved space (particularly on a sphere). To examine the efficiency of this scheme, it has been applied to the exactly solvable Kepler–Coulomb problem and its quasi-exactly solvable rational extensions. The highly accurate values of the energies corresponding to some of the hidden bound states and the approximate Shannon information entropies of the first five bound states of the above-mentioned potentials have been presented here. Finally, the findings of our results have been summarized in the concluding section.</p>