While it is known that unconditionally secure position-based cryptography is
impossible both in the classical and the quantum setting, it has been shown
that some quantum protocols for position verification are secure against
attackers which share a quantum state of bounded dimension. In this work, we
consider the security of two protocols for quantum position verification that
combine a single qubit with classical strings of total length $2n$: The qubit
routing protocol, where the classical information prescribes the qubit’s
destination, and a variant of the BB84-protocol for position verification,
where the classical information prescribes in which basis the qubit should be
measured. We show that either protocol is secure for a randomly chosen function
if each of the attackers holds at most $n/2 – 5$ qubits. With this, we show for
the first time that there exists a quantum position verification protocol where
the ratio between the quantum resources an honest prover needs and the quantum
resources the attackers need to break the protocol is unbounded. The verifiers
need only increase the amount of classical resources to force the attackers to
use more quantum resources. Concrete efficient functions for both protocols are
also given — at the expense of a weaker but still unbounded ratio of quantum
resources for successful attackers. Finally, we show that both protocols are
robust with respect to noise, making them appealing for applications.

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