  In Private Broadcasting, a single plaintext is broadcast to multiple
recipients in an encrypted form, such that each recipient can decrypt locally.
When the message is classical, a straightforward solution is to encrypt the
plaintext with a single key shared among all parties, and to send to each
recipient a copy of the ciphertext. Surprisingly, the analogous method is
insufficient in the case where the message is quantum (i.e. in Quantum Private
Broadcasting (QPB)). In this work, we give three solutions to $t$-recipient
Quantum Private Broadcasting ($t$-QPB) and compare them in terms of key
lengths. The first method is the independent encryption with the quantum
one-time pad, which requires a key linear in the number of recipients, $t$. We
show that the key length can be decreased to be logarithmic in $t$ by using
unitary $t$-designs. Our main contribution is to show that this can be improved
to a key length that is logarithmic in the dimension of the symmetric subspace,
using a new concept that we define of symmetric unitary $t$-designs, that may
be of independent interest.
