The relative global stability of N-component high-entropy alloys (HEA) in A1, A2, A3 phases are predicted from the formation enthapies using first-principles KKR electronic-structure methods combined with the coherent potential approximation (CPA) to handle chemical and magnetic disorder. In addition, the chemical short-range order and incipient long-range order, dictated by N(N–1)/2 chemical correlations in the HEA, are directly predicted by thermodynamic linear-response theory using the HEA solid-solution phase as a reference state. We exemplify the accurate predictions in various solid-solutions, including AlxCrFeCoNi(1-x), well studied experimentally. We discuss how to use these methods to narrow the design search space for improved HEAs. The method also helps facilitate development of empirical potentials for simulations to predict mechanical behavior to connect to experiment. Research supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for DOE by ISU under Contract DE-AC02- 07CH11358.