The pomegranates are not only linked to Persephone but also to Aphrodite and Hera.
The pomegranate was sacred to Aphrodite who was said to have first planted it on the island of Kypros. The fruit symbolised female fertility and the marital loss of virginity with the consumation of marriage (from both its red stains and rich seeds).
Source: Theoi Greek Mythology
In the ancient & medieval worlds, pomegranates symbolized birth & death, being itself capable of bleeding. It was frequently associated with maidens & maiden-goddesses …. It had simultaneous positive & negative associations, as cthonic divinities, & in particular Earth mothers, were the sources of Life & of Death, of Health & of Illness, of Fertility & Infertility.
Pomegranate as Forbidden Desire was also a component of its meaning within Athena’s Parthenon, but the forbidden desire to eat of the fruit also underlays the Persephone myth & numerous Pomegranate legends.
In Jewish lore it was again the fruit of things forbidden, growing upon the Tree of Knowledge (of sexuality & death) forbidden to Adam & Eve*.
* It depends on the version. For some people, the forbidden fruit is an apple or a pomegranate.