  We address the problem of testing the quantumness of two-dimensional systems
in the prepare-and-measure (PM) scenario, using a large number of preparations
and a large number of measurement settings, with binary outcome measurements.
In this scenario, we introduce constants, which we relate to the Grothendieck
constant of order 3. We associate them with the white noise resistance of the
prepared qubits and to the critical detection efficiency of the measurements
performed. Large-scale numerical tools are used to bound the constants. This
allows us to obtain new bounds on the minimum detection efficiency that a setup
with 70 preparations and 70 measurement settings can tolerate.
