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Self-testing is a powerful certification of quantum systems relying onmeasured, classical statistics. This paper considers self-testing in bipartiteBell scenarios with small number of inputs and outputs, but with quantum statesand measurements of arbitrarily large dimension. The contributions are twofold.Firstly, it is shown that every maximally entangled state can be self-testedwith four binary measurements per party. This result extends the earlier workof Man\v{c}inska-Prakash-Schafhauser (2021), which applies to maximallyentangled states of odd dimensions only. Secondly, it is shown that everysingle binary projective measurement can be self-tested with five binarymeasurements per party. A similar statement holds for self-testing ofprojective measurements with more than two outputs. These results are enabledby the representation theory of quadruples of projections that add to a scalarmultiple of the identity. Structure of irreducible representations, analysis oftheir spectral features and post-hoc self-testing are the primary methods forconstructing the new self-tests with small number of inputs and outputs.

Constant-sized self-tests for maximally entangled states and single projective measurements