Establish lower bounds for multi-vector VSP

Determine the classical communication complexity of the k-Vector-in-Subspace problem in which Alice receives k independent unit vectors sampled from either the sphere of a Haar-random half-dimensional subspace or its orthogonal complement, for every k≥2.

Background

The paper defines k-VSP as the natural multi-copy communication analogue of Vector-in-Subspace. The cryptographic construction does not require this problem because security for multiple challenge copies follows by a hybrid reduction from the single-copy statement. Nevertheless, the direct communication problem remains unresolved, and the paper reports no lower bound for any k≥2.