  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.
