  Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic
interactions between a classical client and a quantum server. Typically, a
protocol has the quantum server prepare a superposition of two-bit strings of a
claw and then measure it using Pauli-$X$ or $Z$ measurements. In this paper, we
demonstrate a new technique that uses the entire range of qubit measurements
from the $XY$-plane. We show the advantage of this approach in two
applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we
show an optimized two-round proof of quantumness whose security can be
expressed directly in terms of the hardness of the LWE (learning with errors)
problem. Second, we construct a one-round protocol for blind remote preparation
of an arbitrary state on the $XY$-plane up to a Pauli-$Z$ correction.
