Do Mormon women hold the priesthood? For the majority of Latter-day Saints, the answer is an obvious no: women do not hold the priesthood. For me, the answer is an intellectually frustrating “maybe, yes, no, dunno.” In part, I have found Quinn’s research compelling: clearly there was some sense in the 19th century that temple-endowed Mormon women were part of the “priesthood” (see also J. Stapley’s thoughts). There’s also the rather practical realisation that if Mormon women can dress in the robes of the priesthood, wear the priesthood garment, and enact the rites of the priesthood, they are quite obviously “priests” (see Compton).
On the other hand, it is not clear to me that the 19th century use of the term “priesthood” has an equivalent in modern Mormonism (let alone in wider religious theory). Just because Mormon women could be members of an Anointed Quorum, for example, does not mean that they held the “priesthood” in any sense that would be meaningful for us today. Also, it’s rather obvious that priesthood or no, women are not ordained to priesthood office. For me, I believe there is “a priesthood” available to Mormon women, but I don’t quite know where to situate it and we would need an authorised clarification in order to better understand it.
One thing Quinn has written particularly catches my eye. He suggests that the lack of ordination of women, and the separation of Mormon womens’ “priesthood” from the bureaucratic function of the church may, in fact, be a useful thing:
A church president continued to affirm the role of women as prophetesses into the twentieth century. “I believe that every mother has the right to be a prophetess and to have the gift of sight, foreseeing prescience, to foresee danger and evil and to know what to do in her family and in her sphere,” Joseph F. Smith affirmed in 1913. “They are prophetesses, they are seers, they are revelators to their households and to their families . . .”Without ordination to specific offices of priesthood, women have avoided aspirations and abuses common to church offices reserved for men (D&C 121:34-40) [emphasis mine].
According to this latter formulation by Quinn, priesthood office, whilst an important cog in the machinery of the kingdom, affords power to the holder and therefore runs the risk of being abused. Priesthood office means that men often serve busily in the priesthood bureaucracy of the church and may find themselves tempted by the power of position (a situation D&C 121 finds inevitable). Less encumbered by this risk, a woman may be free to be a “prophetess” in what is now mostly an organisation of “priests.”(And not just “in her family,” I would say.) 
I would like to make two things clear. First, whilst I am attracted to this idea, I would not wish to use it as an apologia for why women should not hold priesthood office. It is a potential benefit that arises from (but is not the basis of) the status quo. Second, I realise that “woman-as-prophet” is not a natural notion in the church today, so this may be an unfulfilled opportunity and one difficult to realise. (And a third, I guess: not all men with priesthood office are power-hungry autocrats! And of course, some women are exactly that. Women abuse leadership positions in the church too. But they cannot fall on their priesthood office to justify it.)
Still, I feel that this notion offers something for those who would like to raise the status of women in the church. We can even disagree entirely with Quinn’s idea, but still use it to remind us that “prophet(ess)” is not a priesthood office, and that lack of office need not mean a lack of women’s voices and a void in the application of women’s gifts for the benefit of the whole body of Christ. In fact, according to Quinn, lack of office may be what ought to bring these gifts to the fore, if we would let them.
Whilst questions of women’s priesthood should continue to be explored, it must also be accepted that any radical movement in the church can only be enacted by people whose authority is greater than our own. Until then, it may be a dead-end. But we can, at the level of family and ward, better allow women to exercise their gifts. I think there is an opportunity (unfulfilled?) for women to be prophets in the appropriate way. I invite your suggestions as to how women can fill this apposite role.
1. The idea of priest vs. prophet is probably rather simplistic but remains useful nevertheless. It maintains that spiritual movements often begin through the actions of the “prophet,” full of charisma and often an offender of orthodoxy. As spiritual movements become religions, the “priest” then comes to govern the ecclesiastical structure that coalesces around the prophet’s teachings. Over time, religions tend to be priestly rather than prophetic.
2. Shall we define “prophet”? How about: 1. Someone with a testimony of Jesus, and who finds the opportunity to share that witness 2. Someone with the gift of “prophecy,” or special insight into her own spiritual affairs and those of others. An appendage to these might be the exercise of the other gifts of the spirit that a woman is entitled to enjoy. And yes, I realise that the question of spheres of authority makes all of this rather tricky. But there you go.