[2024-12-27 Weiyuan Gong] We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate tr(ρ2) of an unknown quantum state ρ to additive error ϵ. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate tr(ρσ) to additive error ϵ given copies of unknown quantum state ρ and σ using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with k-qubit one-way quantum communication and unentangled local measurements using O(median{1/ϵ2,2n/2/ϵ,2n−k/ϵ2}) copies of ρ and σ. Our protocol can be modified to estimate the purity of an unknown quantum state ρ using k-qubit quantum memory with the same complexity. We prove that arbitrary protocols with k-qubit quantum memory that estimate purity to error ϵ require Ω(median{1/ϵ2,2n/2/ϵ√,2n−k/ϵ2}) copies of ρ. This indicates the same lower bound for quantum inner product estimation with one-way k-qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to Ω(max{1/ϵ2,2n/2/ϵ}) for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.