  A $((k,n))$ quantum threshold secret sharing (QTS) scheme is a quantum
cryptographic protocol for sharing a quantum secret among $n$ parties such that
the secret can be recovered by any $k$ or more parties while $k-1$ or fewer
parties have no information about the secret. Despite extensive research on
these schemes, there has been very little study on optimizing the quantum
communication cost during recovery. Recently, we initiated the study of
communication efficient quantum threshold secret sharing (CE-QTS) schemes.
These schemes reduce the communication complexity in QTS schemes by accessing
$d\geq k$ parties for recovery; here $d$ is fixed ahead of encoding the secret.
In contrast to the standard QTS schemes which require $k$ qudits for recovering
each qudit in the secret, these schemes have a lower communication cost of
$\frac{d}{d-k+1}$ for $d>k$. In this paper, we further develop the theory of
communication efficient quantum threshold schemes. Here, we propose universal
CE-QTS schemes which reduce the communication cost for all $d\geq k$
simultaneously. We provide a framework based on ramp quantum secret sharing to
construct CE-QTS and universal CE-QTS schemes. We give another construction for
universal CE-QTS schemes based on Staircase codes. We derived a lower bound on
communication complexity and show that our constructions are optimal. Finally,
an information theoretic model is developed to analyse CE-QTS schemes and the
lower bound on communication complexity is proved again using this model.
