  Unsharp quantum measurements provide a resource in scenarios where one faces
the trade-off between information gain and disturbance. In this work we
introduce a prepare-transform-measure scenario in which two-outcome unsharp
measurements outperform their sharp counterparts, as well as any stochastic
strategy involving dichotomic projective measurements. Based on that, we
propose a scheme for semi-device-independent self-testing of unsharp
measurements and show that all two-outcome qubit measurements can be
characterized in a robust way. Along with the main result, in this work we
introduce a method, based on semidefinite programming, for bounding quantum
correlations in scenarios with sequential measurements of length two. This
method can also be applied to refine security analysis of the
semi-device-independent one-way quantum key distribution. We also present new
information gain-disturbance relation for pairs of dichotomic measurements.
