Presently, multiple providers give access to their NISQ (Noisy Intermediate-Scale Quantum) computers. The computational performance may be enhanced by error detection and correction. Another key question at this stage is how to benchmark these quantum computers. In the framework of the Quantum Information National Laboratory, we aim at studying different types of errors and noise inherent in quantum computers. In order to assess gate errors, we develop general-purpose randomized benchmark schemes built on unitary designs, which go beyond the popular Clifford gates. Another prospective benchmark tool is based on measurement-induced nonlinear dynamics of iterated quantum protocols, the theory of which we have been developing recently. These protocols are appropriate for test purposes, as the dynamics can be extremely sensitive to noise.