Pearl, now seven, begins to display the quality of intuition. Give a few examples.

I think Pear begins to display the quality of intuition
even before she is 7.  It starts in Chapter 6 - when Pearl seems to only recognize her
mother because of the scarlet letter.  As a baby and a small child it is referenced that
she is frequently touching it and responding to it, almost as if she knows and
understands its power over her mother.

As she gets older,
she starts to show signs of understanding things about people in the rest of the
colony.  She is unafraid to chase the children to attempt to throw rocks at her and her
mother as they walk through town.  When asked at the Governor's house (loosely quoting
here) "Where did you come from/Who made you?" (this said to see if Hester was raising
her child in proper catechism), Pearl responds with something to the effect of: "I was
not born at all.  I was plucked from the rose bush that grows beside the prison door." 
What a response from a 7 year old.

It is also right about
this time that Pearl begins to repeatedly question her mother about the scarlet letter. 
And this question always comes coupled with another, (again, loosely quoting) "Why do
you [Hester] wear the scarlet letter on your breast" and "Why does the minister hold his
hand over his heart?"  She also asks why the minister is always seen with
Chillingworth.

Rather than coming across as innocent
questions, these seem meant to be rhetorical.  It is as if only Pearl recognizes the
answers the lie in these questions, and by asking them, she is attempting to reveal the
answers, rather than her curiosity.