  Incompatibility of certain measurements -- impossibility of obtaining
deterministic outcomes simultaneously -- is a well known property of quantum
mechanics. This feature can be utilized in many contexts, ranging from Bell
inequalities to device dependent QKD protocols. Typically, in these
applications the measurements are chosen from a predetermined set based on a
classical random variable. One can naturally ask, whether the non-determinism
of the outcomes is due to intrinsic hiding property of quantum mechanics, or
rather by the fact that classical, incoherent information entered the system
via the choice of the measurement. Authors of [NJP 19, 023038 (2017)] examined
this question for a specific case of two mutually unbiased measurements on
systems of different dimensions. They have somewhat surprisingly shown that in
case of qubits, if the measurements are chosen coherently with the use of a
controlled unitary, outcomes of both measurements can be guessed
deterministically. Here we extend their analysis and show that specifically for
qubits, measurement result for any set of measurements with any a-priori
probability distribution can be faithfully guessed by a suitable state
preparation and measurement. We also show that up to a small set of specific
cases, this is not possible for higher dimensions. This result manifests a deep
difference in properties of qubits and higher dimensional systems and suggests
that these systems might offer higher security in specific cryptographic
protocols. More fundamentally, the results show that the impossibility of
predicting a result of a measurement is not caused solely by a loss of
coherence between the choice of the measurement and the guessing procedure.
