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This paper unifiedly addresses two kinds of key quantum secure tasks, i.e.,quantum versions of secret sharing (SS) and symmetric private informationretrieval (SPIR) by using multi-target monotone span program (MMSP), whichcharacterizes the classical linear protocols of SS and SPIR. SS has two quantumextensions; One is the classical-quantum (CQ) setting, in which the secret tobe sent is classical information and the shares are quantum systems. The otheris the quantum-quantum (QQ) setting, in which the secret to be sent is aquantum state and the shares are quantum systems. The relation between thesequantum protocols and MMSP has not been studied sufficiently. We newlyintroduce the third setting, i.e., the entanglement-assisted (EA) setting,which is defined by modifying the CQ setting with allowing prior entanglementbetween the dealer and the end-user who recovers the secret by collecting theshares. Showing that the linear version of SS with the EA setting is directlylinked to MMSP, we characterize linear quantum versions of SS with the CQ ad QQsettings via MMSP. Further, we introduce the EA setting of SPIR, which is shownto link to MMSP. In addition, we discuss the quantum version of maximumdistance separable codes.

Unified Approach to Secret Sharing and Symmetric Private Information Retrieval with Colluding Servers in Quantum Systems