The classical Borsuk-Ulam theorem may be seen as a statement about the complexity of the spheres as principal $\mathbb{Z}/2\mathbb{Z}$-bundles via the antipodal action. The truthfulness of analogous statements in the noncommutative setting for general compact groups, or even compact quantum groups, were proposed by Baum, Dąbrowski, and Hajac. For classical principal bundles, a dimensional notion called $G$-index plays a crucial role in this quest. I will talk about my joint work with Gardella, Hajac, and Tobolski, where we introduce the local triviality dimension, a generalization of $G$-index for noncommutative principal bundles that can be used to transfer Borsuk-Ulam-type results from the classical setting to the noncommutative setting.