Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, a broad class of topological states, namely, symmetry-protected topological (SPT) states, can be identified by distinctive degeneracies in their entanglement spectrum. Here, we propose and realize two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The two protocols link quantum information processing to the classification of SPT phases of matter. They invoke the creation of a cluster state and are implemented on an IBM quantum computer. The experimental findings are compared to noisy simulations, allowing us to study the stability of topological states to perturbations and noise. *We acknowledge support from ARO (W911NF-20-1-0013) (R. R. and E. S.). This work is supported by the Israel Science Foundation, Grants No. 151/19 (D. A. and E. G. D. T.) and No. 154/19 (D. A., Y. N., E. S., and E. G. D. T.).