Effective two-body potentials for electron-ion plasmas are analyzed. Such potentials which have been previously derived by Kelbg and others capture the basic quantum diffraction effects and are exact in the weak coupling limit. Moreover, using path integral Monte Carlo methods, they can be applied to strongly coupled plasmas which include bound states. We investigate the accuracy of the diagonal Kelbg potential as well as the off-diagonal Kelbg potential by comparison with accurate numerical solutions of the off-diagonal two-particle Bloch equation. Results for the energies and pair distribution functions are presented. A significant improvement is achieved by correcting the potential value at zero particle separation, and analytical expressions for improved interaction potential between electrons and protons and electrons with parallel and antiparallel spins are obtained. Further, application of the quantum pair potentials in path integral Monte Carlo and in classical molecular dynamics simulations of dense hydrogen are presented. Finally, we point out an interesting relation between the quantum potential and effective potentials used in density functional theory.