The most common and most likely answer to this crossword clue is the 12 letter word VENERABILITY
I've seen this crossword clue in The Daily Telegraph
V
E
N
E
R
A
B
I
L
I
T
Y
Buddy explains!
The straight part
We know that "Presumed quality of archdeacon" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case presumed quality of archdeacon = venerability.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.