We report the successful synthesis of single-crystalline cuprate superconductors HgBa$_{2}$CaCu$_{2}$O$_{6+delta}$ and HgBa$_{2}$Ca$_{2}$Cu$_{3}$O$_{8+delta}$. These compounds are well-known for their high optimal superconducting critical temperatures of $T_mathrm{c}$ = 128 K and 134 K at ambient pressure, respectively, and for their challenging synthesis. Using a conventional quartz-tube encapsulation method and a two-layer encapsulation method that utilizes custom-built high-pressure furnaces, we are able to grow single crystals with linear dimensions up to several millimeters parallel to the CuO$_2$ planes. Extended post-growth anneals are shown to lead to sharp superconducting transitions, indicative of high macroscopic homogeneity. X-ray diffraction and polarized Raman spectroscopy are identified as viable methods to resolve the seemingly unavoidable inter-growth of the two compounds. Our work helps to remove obstacles toward the study of these model cuprate systems with experimental probes that require sizable high-quality crystals.
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