  The relationship between correlations and entanglement has played a major
role in understanding quantum theory since the work of Einstein, Podolsky and
Rosen (1935). Tsirelson (1980) proved that Bell states, shared among two
parties, when measured suitably, achieve the maximum non-local correlations
allowed by quantum mechanics. Conversely, Reichardt, Unger and Vazirani (2013)
showed that observing the maximal correlation value over a sequence of repeated
measurements, implies that the underlying quantum state is close to a tensor
product of maximally entangled states and, moreover, that it is measured
according to an ideal strategy. However, this strong rigidity result comes at a
high price, requiring a large number of entangled pairs to be tested. In this
paper, we present a significant improvement in terms of the overhead by instead
considering quantum steering where the device of the one side is trusted. We
first demonstrate a robust one-sided device-independent version of
self-testing, which characterises the shared state and measurement operators of
two parties up to a certain bound. We show that this bound is optimal up to
constant factors and we generalise the results for the most general attacks.
This leads us to a rigidity theorem for maximal steering correlations. As a key
application we give a one-sided device-independent protocol for verifiable
delegated quantum computation, and compare it to other existing protocols, to
highlight the cost of trust assumptions. Finally, we show that under reasonable
assumptions, the states shared in order to run a certain type of verification
protocol must be unitarily equivalent to perfect Bell states.
