Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can restore the original secret from their shares. When we encode a secret by a secret sharing scheme and distribute shares, sometimes not all participants are accessible, and it is desirable to distribute shares to those participants before a secret information is determined. Secret sharing schemes for classical secrets have been known to be able to distribute some shares before a given secret. Lie et al. found a ((2, 3))-threshold secret sharing for quantum secrets can distribute some shares before a given secret. However, it is unknown whether distributing some shares before a given secret is possible with other access structures of secret sharing for quantum secrets. We propose a quantum secret sharing scheme for quantum secrets that can distribute some shares before a given secret with other access structures.

Stabilizer-based quantum secret sharing has two methods to reconstruct a quantum secret: The erasure correcting procedure and the unitary procedure. It is known that the unitary procedure has a smaller circuit width. On the other hand, it is unknown which method has smaller depth and fewer circuit gates. In this letter, it is shown that the unitary procedure has smaller depth and fewer circuit gates than the erasure correcting procedure which follows a standard framework performing measurements and unitary operators according to the measurements outcomes, when the circuits are designed for quantum secret sharing using the [[5, 1, 3]] binary stabilizer code. The evaluation can be reversed if one discovers a better circuit for the erasure correcting procedure which does not follow the standard framework.

We show a construction of a quantum ramp secret sharing scheme from a nested pair of linear codes. Necessary and sufficient conditions for qualified sets and forbidden sets are given in terms of combinatorial properties of nested linear codes. An algebraic geometric construction for quantum secret sharing is also given.

The multiple assignment scheme is to assign one or more shares to single participant so that any kind of access structure can be realized by classical secret sharing schemes. We propose its quantum version including ramp secret sharing schemes. Then we propose an integer optimization approach to minimize the average share size.

We introduce a graphical calculus for multi-qutrit systems (the qutrit ZX-calculus) based on the framework of dagger symmetric monoidal categories. This graphical calculus consists of generators for building diagrams and rules for transforming diagrams, which is obviously different from the qubit ZX-calculus. As an application of the qutrit ZX-calculus, we give a graphical description of a (2, 3) threshold quantum secret sharing scheme. In this way, we prove the correctness of the secret sharing scheme in a intuitively clear manner instead of complicated linear algebraic operations.