The idea of quantum space was proposed long ago, but directly measuring its effects remains far in the future — simply because we cannot yet generate the energies required to prove it. Fortunately, even though the effects of such a structure are minimal, under certain conditions we can make use of the best high-energy generator available: the Universe itself, which regularly produces powerful high-energy bursts. If those bursts originate from distant sources, the tiny effects of quantum space can accumulate over long distances into a measurable magnitude. Many studies in this field rely on perturbative methods, typically considering only first- or second-order corrections. However, this work employs a full dispersion law derived from a three-dimensional quantum space model. Our primary focus is investigating the phenomena of in-vacuo dispersion and the threshold anomaly. We also briefly consider how additional spatial dimensions might influence the behavior of such a quantum space.