High entropy alloy nanoparticles are envisaged as one of the most interesting materials compared to monoatomic materials due to their modulated properties in terms of their convenient surface-to-volume ratio. However, studies are still missing to unveil how composition or nanoparticle size can influence nanoparticle morphology. Based on molecular dynamics simulations, we perform a structural characterization as a function of nanoparticle size and the chemical composition of high entropy alloy nanoparticles subject to multiple annealing cycles. After the multiple thermal loads, we observe a substantial migration of copper atoms towards the np surface, consistent with the experimental results of Cu-based high entropy alloys. The resulting high entropy alloy nanoparticle behaves as a core-shell nanostructure with a rich fcc phase on the surface (50% of Cu) and 5% fcc phase in the nanoparticle core. Inspecting the nanoparticle surface, it is observed that high entropy alloy nanoparticles have a lack of surface facets, leading to a more spherical shape, quite different from mono-metallic nanoparticles with a high number of facets. Performing an average atoms simulation, it showed that nanoparticles are prone to form 111 surface facets independent of the nanoparticle size, suggesting that for high entropy alloy nanoparticles, the chemical complexity avoids the formation of surface facets. The latter can be explained in terms of the lattice distortion inducing tensile/compressive stress that drives the surface reconstruction. All in all our results match extremely well with experimental evidence of FeNiCrCoCu nanocrystalline materials, explaining the Cu segregation in terms of surface energy and mixing enthalpy criteria. We believe that our results provide a detailed characterization of high entropy nanoparticles focusing on how chemical complexity induces morphological changes compared to mono-crystalline nanoparticles. Besides, our findings are valuable for experimental works aimed at designing the shape and composition of multicomponent nanoparticles.