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We introduce a new family of quantum secret sharing protocols with limitedquantum resources which extends the protocols proposed by Markham and Sandersand by Broadbent, Chouha, and Tapp. Parametrized by a graph G and a subset ofits vertices A, the protocol consists in: (i) encoding the quantum secret intothe corresponding graph state by acting on the qubits in A; (ii) use aclassical encoding to ensure the existence of a threshold. These new protocolsrealize ((k,n)) quantum secret sharing i.e., any set of at least k playersamong n can reconstruct the quantum secret, whereas any set of less than kplayers has no information about the secret. In the particular case where thesecret is encoded on all the qubits, we explore the values of k for which thereexists a graph such that the corresponding protocol realizes a ((k,n)) secretsharing. We show that for any threshold k> n-n^{0.68} there exists a graphallowing a ((k,n)) protocol. On the other hand, we prove that for any k<79n/156 there is no graph G allowing a ((k,n)) protocol. As a consequence thereexists n_0 such that the protocols introduced by Markham and Sanders admit nothreshold k when the secret is encoded on all the qubits and n>n_0.

New Protocols and Lower Bound for Quantum Secret Sharing with Graph States