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Short-pulse (<; 10 ps) high-intensity (1019-1021 W·cm-2) laser-target interactions produce high-density plasmas, in which a hot (nonthermal) collisionless electron population is generated via a number of energy absorption mechanisms. A key requirement in many applications of these interactions is to achieve maximum conversion efficiency of laser energy into hot electrons of a specific energy. Measurement of the hot-electron temperature and the associated hot-electron fraction supports the understanding of energy absorption mechanisms present in the 1019-1021 W·cm-2 intensity regime. One highly useful signature of hot-electron generation is the bremsstrahlung radiation emission due to the hot electrons interacting electromagnetically with cold target atoms. We present the design, characterization, and modeling of two target diagnostics to measure the high-energy (100 keV-2 MeV) bremsstrahlung emission from hot electrons in laser-plasma experiments; a thermoluminescent dosimeter (TLD) array and a hard X-ray spectrometer (HXRS). These diagnostics will exploit new techniques to determine the hot-electron distribution generated with the short-pulse beamlines of the upcoming high-power Orion laser system at AWE. Past bremsstrahlung dose measurements obtained with a TLD array (of similar design to the Orion one) are used to demonstrate how the bremsstrahlung production efficiency, an indicator of hot-electron generation, can change with laser parameters, target type, and experimental geometry. In addition, we use results from characterization of the HXRS, a diagnostic which collects channel charges in defined bremsstrahlung spectral ranges, to establish a new method enabling the diagnosis of hot-electron temperatures.