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The exotic region of the nuclear chart around the doubly-magic 78Ni (Z = 28, N = 50) nucleus presents an excellent testing ground for our understanding of nuclear structure. The region around N = 50 plays an important role in the astrophysical rapid neutron- capture (r) process. The robustness of the N = 50 conventional magic number could affect current knowledge of nuclear abundances. Measurements of 2+ and 4+ states offer one of the first indications of shell structure and evolution. Measurements of low-lying states in 81,82,83,84Zn were performed. 82,84Zn are the first two even-even nuclei beyond 78Ni, thus are an important test for the strength of the doubly-magic 78Ni core. The Zn nuclei were measured in the first and second experimental campaigns of the SEASTAR (Shell Evolution and Search for Two-plus energies At the RIBF) project conducted at the RIBF, RIKEN, Japan. In-flight fragmentation-fission beams were produced using a 238U primary beam of 345 MeV/nucleon. The BigRIPS spectrometer was used to identify and select the secondary beams impinged onto a liquid hydrogen target. γ rays were detected using DALI2, an array of 186 NaI detectors surrounding the target. The MINOS system provided vertex reconstruction by tracking the outgoing protons in (p, 2p) proton knockout reactions. Reaction products were identified using the ZeroDegree spectrometer. The 2+ and 4+ states in 82Zn were measured, the 4+1 was observed for the first time in this work along with two additional states. The 2+ and 4+ in 84Zn were measured for the first time, extending the known Zn systematics to N = 54. Two new transitions were observed and assigned within 81Zn, while two tentative transitions were seen in 83Zn. Two state-of-the-art shell model calculations were performed: Ni78-II utilised an inert 78Ni core with a large valence space, while the MCSM calculation allowed core-breaking configurations with a limited neutron valence space. Comparisons of these calculations to the results and region systematics have revealed that the breaking of the 78Ni core provides a significant contribution to low-lying states in the direct vicinity of 78Ni. However, the inclusion of core-breaking configurations also needs to be met with an ample model space to accommodate near future measurements of nuclei in the region. Interactions need to be developed to enable such accommodations in current shell-model calculations.