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The excellent mechanical properties of metallic nanolaminates are closely related to the thickness of each individual layer (h). In this work, we comparatively investigated the effects of h on the mechanical properties of coherent Cu/CoCrFeNi and Cu/Ni nanolaminates under uniaxial compression using molecular dynamics simulations. It showed that with the decrease of h, the mechanical properties of the nanolaminates would be improved significantly. As h is decreased to a few nanometers, the flow stress of Cu/CoCrFeNi nanolaminates ascends to a plateau, while that of Cu/Ni nanolaminates descends. Two deformation mechanisms: dislocations slip constrained within layers and slip across interface, were revealed for the Cu/CoCrFeNi nanolaminates. Furthermore, the underlying mechanisms by which the Cu/CoCrFeNi and Cu/Ni nanolaminates exhibit different flow strength trends are revealed from the perspective of dislocation density. A model extended from the confined layer slip and interface barrier strength model was proposed to quantitatively estimate the flow stress of Cu/CoCrFeNi nanolaminates at different temperatures and strain rates. The results presented would be of great significance for the development and application of Cu/HEA nanolaminates.