I have no idea what efficacy will apply to sacrifices made under the Mosaic law, as it has been fulfilled.
Any future sacrifices in a third temple will have no efficacy.
About 40 years before the temple was destroyed, (which Christians believe it was right after the time of Jesus’ death/resurrection) several signs happened regarding the temple and temple sacrifices. These signs are recorded in Jewish history. 2 very important signs were regarding the Day of Atonement ceremony. Here is one of them that especially stands out: When the two goats were chosen–one to be sacrificed to make atonement and the other as the scapegoat to be set free, a red chord was wrapped around then scapegoat’s neck in order to tell the two goats apart because they had to be identical. The sins were laid on the scapegoat and the “innocent” goat was sacrificed. After the sacrifice the red chord would “miraculously” turn white signifying that God had accepted their sacrifice. (“Though your sins are like scarlet, I will make them white as snow”) But somewhere after 30 a.d. the red chord no longer turned white–signifying that God no longer accepted this sacrifice because Jesus’ sacrifice had fulfilled the requirement. (That’s the explanation Christians would give.)
There were also 3 other signs. I’ll just copy the signs here and note a website for reference.
‘Forty years before the destruction of the Temple,’ says the Jerusalem Talmud, ‘the western light went out, the crimson thread remained crimson, and the lot for the Lord always came up in the left hand. They would close the gates of the Temple by night and get up in the morning and find them wide open’ (
The Yerushalmi , translated by Jacob Neusner, p.156f).
According to the Babylonian Talmud, ‘Our Rabbis taught: During the last forty years before the destruction of the Temple the lot [‘for the Lord’] did not come up in the right hand; nor did the crimson-colored strap become white; nor did the western-most light shine; and the doors of the
Hekal (Temple) would open by themselves’ (The
Soncino Talmud , tractate ‘Yoma,’ 39b).
Here’s this particular website:
Miracles of Yom Kippur – Etz Chayim Messianic Synagogue