"The thing at issue is buried intentions-the secret allegiances of the alienated heart, always the main threat to the theocratic mind, as well as its immemorial quarry"
How does this relate to "The Crucible"?
You can relate this best if you break it down into parts. The first part mentions "buried intentions", perhaps suggesting secret intentions or purposes; next, the quote explains that those buried intentions are "the secret allegiances of the alienated heart". Allegiance means 'loyalty', so in this case the secret intentions stems from the loyalty of the alienated heart. I would think this would refer to Abigail, in that her love is being pushed away, or alienated, by John Proctor, and this action is the secret motivator of her intentions behind the accusations.
"[A]lways the threat to the theocratic mind" could suggest that this pushing away of love, or this emotional pain, could be the biggest threat to this way of living, in that their simple, theocratic society leaves no room for passion.
The last part of the quote is the hardest to understand. "[I]mmemorial quarry" is unclear in terms of what it is to whcih it refers. An immemorial quarry would mean something that is abundant and is beyond memory, stretching far back in history. It could refer to the theocracy in which they live.
Hope this helps! Good luck!
The playwright, Arthur Miller, is trying to make the point that the motives for accusing someone of witchcraft were not always pure. Many people of Salem had nursed grudges and maintained feuds with their neighbors and others of the community. Since they couldn't openly fight against their neighbors, for that would have been in violation of the Biblical injunction to "love your neighbor as yourself", they discovered they could accuse their neighbors of practicing witchcraft. Thus they could secretly get revenge for other things under the guise of searching out sin. In other words, they, like many others through the ages, had hidden motives and agendas. This idea also reflects the sometimes devious and evil motives of the accusers during the McCarthy era, which the Crucible parallels.