Soft gamma repeaters and anomalous X-ray pulsars are subset of slow-rotating neutron stars, known as magnetars, that have extremely high inferred surface magnetic fields, of the order 100-1000 TeraGauss. Hard, non-thermal and pulsed persistent X-ray emission extending between 10 keV and 230 keV has been seen in a number of magnetars by RXTE, INTEGRAL, and Suzaku. In this thesis, the author considers inner magnetospheric models of such persistent hard X-ray emission where resonant Compton upscattering of soft thermal photons is anticipated to be the most efficient radiative process. This high efficiency is due to the relative proximity of the surface thermal photons, and also because the scattering becomes resonant at the cyclotron frequency. At the cyclotron resonance, the effective cross section exceeds the classical Thomson one by over two orders of magnitude, thereby enhancing the efficiency of continuum production and cooling of relativistic electrons. In this thesis, a new Sokolov and Ternov formulation of the QED Compton scattering cross section for strong magnetic fields is employed in electron cooling and emission spectra calculations. This formalism is formally correct for treating spin-dependent effects and decay rates that are important at the cyclotron resonance. The author presents electron cooling rates at arbitrary interaction points in a magnetosphere using the QED cross sections. The QED effects reduce the rates below high-field extrapolations of older magnetic Thomson results. The author also computes angle-dependent upscattering model spectra, formed using collisional integrals, for uncooled monoenergetic relativistic electrons injected in inner regions of pulsar magnetospheres. These spectra are integrated over closed field lines and obtained for different observing perspectives. The spectral cut-off energies are critically dependent on the observer viewing angles and electron Lorentz factor. It is found that electrons with energies less than around 15 MeV will emit most of their radiation below 250 keV, consistent with the observed turnovers in magnetar hard X-ray tails. Moreover, electrons of higher energy still emit most of the radiation below 1 MeV, except for very select viewing perspectives that sample tangents to field lines. This small parameter space makes it difficult to observe signals extending into the Fermi-LAT band. Polarization dependence in spectra is illustrated, offering potential constraints for models of magnetar emission in anticipation of a future hard X-ray polarimetry missions.