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In one-out-of-$m$ spacetime-constrained oblivious transfer (SCOT), Alice andBob 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 isMinkowski, or close to Minkowski; Alice inputs a message $x_i$ in the causalpast of a spacetime point $Q_i$ of $R_i$, for $i\in\{0,1,\ldots,m-1\}$; Bobinputs $b\in\{0,1,\ldots,m-1\}$ 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 $i,j\in\{0,1,\ldots,m-1\}$. Weintroduce unconditionally secure one-out-of-$m$ SCOT protocols extending theone-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 asubroutine to implement a class $\mathcal{P}_{\text{CC}}$ of one-out-of-$m$SCOT protocols where distant agents only need to communicate classically. Wepresent unconditionally secure one-out-of-$m$ DQACM protocols andone-out-of-$m$ SCOT protocols of the class $\mathcal{P}_{\text{CC}}$, forarbitrary integers $m\geq2$. We discuss various generalizations of SCOT. Inparticular, we introduce a straightforward extension of SCOT to a$k$-out-of-$m$ setting, and suggest protocols where distant agents only need tocommunicate classically, while we leave the investigation of their security asan open problem.

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