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In this work, the deformation behavior of as-prepared (AP) and structurally relaxed (SR) Cu–Zr–based nanoglasses (NGs) are investigated using nano- and micro-indentation. The NGs are subjected to structural relaxation by annealing them close to the glass transition temperature without altering their amorphous nature. The indentation load, p, vs. displacement, h, curves of SR samples are characterized by discrete displacement bursts, while the AP samples do not show any of them, suggesting that annealing has caused a local change in the amorphous structure. In both the samples, hardness (at nano- and micro-indentation) decreases with increasing p, demonstrating the indentation size effect. The micro-indentation imprints of SR NGs show evidence of shear bands at the periphery, indicating a heterogeneous plastic flow, while AP NG does not display any shear bands. Interestingly, the shear band density decreases with p, highlighting the fact that plastic strain is accommodated entirely by the shear bands in the subsurface deformation zone. The results are explained by the differences in the amorphous structure of the two NGs.