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Regarding women saying Kiddush Levana, the Mishnah Berurah Siman 426 סק"א, quoting the Shlah, writes:
ונשים פטורות מלקדש הלבנה דהוי מ"ע שהזמן גרמא ואף דכל מ"ע שהזמן גרמא נוהגות הנשים שמקיימות ומברכות עליהן מ"מ מצוה זו אין צריכין לקיימה משום דהם גרמו פגם הלבנה.
Women are exempt from blessing the moon, since it's a positive time-bound mitzvah. Even though women do practice positive time-bound mitzvos, even making blessings on them, nevertheless this mitzvah they don't need to fulfill, since they caused the Moon's blemish.
1) What does it mean women caused the Moon's blemish? The only thing I can think of is Chava tempting Adam to eat from the tree somehow caused the moon to shrink, but Chazal in Chulin 60b say that happened earlier in creation. Unless you say that the moon will be restored to its glory when Moshiach comes, when the world's purpose comes to fruition, which would have happened the first week of creation had Chava not sinned...
2) If women cause the blemish, shouldn't they help fix it by blessing the moon and praying for its revival? Similar to what the Mishnah Berurah Siman 263 סקי"ב writes regarding why specifically women light Shabbos candles:
ועוד טעם מפני שכבתה נרו של עולם שגרמה מיתה לאדם הראשון
Another reason is because she extinguished the light of the world since she caused the death of Adam