We do not investigate [the genealogy of a bas Kohen]

neither from the Altar and up (i.e. if we find an ancestor that served at the Altar, we need not go any further back),

nor from the duchan (the platform in the Beis Hamikdosh from which a Levi would sing) and up,

nor from a sanhedrin (a beis din of 23 judges that hears capital cases) and up,

and all whose fathers were established to be

from the public officers (who heard monetary cases in Jerusalem)

or a charity administrator

may marry a Kohen

and it is not necessary to investigate [their genealogy] any further.

Rabbi Yosi says

also someone who had signed as a witness

in the court of Yeshanah near Tzipori

Rabbi Chanina ben Antignos says

also someone who was listed

in the list of military officers of King [David].

The daughter of a male Challal (someone born from a Kohen's forbidden relationship)

is disqualified from [marrying] a Kohen forever (i.e. no matter how many generations removed from the original Challal).

A Yisroel that marries a Challalah,

his daughter is permitted to a Kohen.

A Challal that marries a bas Yisroel,

his daughter is disqualified for Kohen.

Rabbi Yehuda says

the daughter of a male convert

is like the daughter of a male Challal.