One-out-of-$m$ spacetime-constrained oblivious transfer

In one-out-of-$m$ spacetime-constrained oblivious transfer (SCOT), Alice and Bob agree on $m$ pairwise spacelike separated output spacetime regions $R_0,R_1,\ldots, R_{m-1}$ in an agreed reference frame in a spacetime that is Minkowski, or close to Minkowski; Alice inputs a message $x_i$ in the causal past of a spacetime point $Q_i$ of $R_i$, for $i\in\{0,1,\ldots,m-1\}$; Bob inputs $b\in\{0,1,\ldots,m-1\}$ in the intersection of the causal pasts of $Q_0,Q_1,\ldots,Q_{m-1}$ and outputs $x_b$ in $R_b$; Alice remains oblivious to $b$ anywhere in spacetime; and Bob is unable to obtain $x_i$ in $R_i$ and $x_j$ in $R_j$ for any pair of different numbers $i,j\in\{0,1,\ldots,m-1\}$. We introduce unconditionally secure one-out-of-$m$ SCOT protocols extending the one-out-of-two SCOT protocols of Pital\'ua-Garc\'ia [Phy. Rev. A 93, 062346 (2016)] and Pital\'ua-Garc\'ia and Kerenidis [Phy. Rev. A 98, 032327 (2018)], for arbitrary integers $m\geq 2$. We define the task of one-out-of-$m$ distributed quantum access with classical memory (DQACM), which works as a subroutine to implement a class $\mathcal{P}_{\text{CC}}$ of one-out-of-$m$ SCOT protocols where distant agents only need to communicate classically. We present unconditionally secure one-out-of-$m$ DQACM protocols and one-out-of-$m$ SCOT protocols of the class $\mathcal{P}_{\text{CC}}$, for arbitrary integers $m\geq2$. We discuss various generalizations of SCOT. In particular, we introduce a straightforward extension of SCOT to a $k$-out-of-$m$ setting, and suggest protocols where distant agents only need to communicate classically, while we leave the investigation of their security as an open problem.