The classical Borsuk-Ulam theorem may be seen as a statement about the complexity of the n-spheres as principal Z/2Z-bundles. The truthfulness of analogous statements in the noncommutative setting for general compact groups, or even compact quantum groups, were proposed by Baum, Dabrowski, 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.