  We consider the related notions of two-prover and of relativistic commitment
schemes. In recent work, Lunghi et al. proposed a new relativistic commitment
scheme with a multi-round sustain phase that enables to keep the binding
property alive as long as the sustain phase is running. They prove security of
their scheme against classical attacks; however, the proven bound on the error
parameter is very weak: it blows up doubly exponentially in the number of
rounds. In this work, we give a new analysis of the multi-round scheme of
Lunghi et al., and we show a linear growth of the error parameter instead (also
considering classical attacks only). Our analysis is based on a new and rather
general composition theorem for two-prover commitment schemes. The proof of our
composition theorem is based on a better understanding of the binding property
of two-prover commitments that we provide in the form of new definitions and
relations among them. These new insights are certainly of independent interest
and are likely to be useful in other contexts as well. Finally, our work gives
rise to several interesting open problems, for instance extending our results
to the quantum setting, where the dishonest provers are allowed to perform
measurements on an entangled quantum state in order to try to break the binding
property.
