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 qubits 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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