Quantum Protocols within Spekkens' Toy Model

Quantum mechanics is known to provide significant improvements in information processing tasks when compared to classical models. These advantages range from computational speeds-up to security improvements. A key question is where these advantages come from. The toy model developed by Spekkens [R. W. Spekkens PRA 75, 032110 (2012)] mimics many of the features of quantum mechanics, such as entanglement and no-cloning, regarded as being important in this regard, despite being a local hidden variable theory. In this work we study several protocols within Spekkens' toy model where we see it can also mimic the advantages and limitations shown in the quantum case. We first provide explicit proofs for the impossibility of toy bit-commitment and the existence of a toy error correction protocol and consequent $k-$threshold secret sharing. Then, defining a toy computational model based on the quantum one-way computer we prove the existence of blind and verified protocols. Importantly, these two last quantum protocols are known to achieve a better-than-classical security. Our results suggest that such quantum improvements need not arise from any Bell-type non locality or contextuality, but rather as a consequence of steering correlations.