Talking about a "cause" for the existence of time is nonsensical since there would need to be the "cause" before there was time, but since there can not be a "before" without time, the very concept breaks apart. Again, "cause" is temporal in nature. It REQUIRES the existence of time.
Time is duration. We measure the passage of time with clocks, and calendars. The Seasons are also evidence of change in the natural environment. Our ageing bodies represent change, measured in years. Human life is finite.
The divine mystery is also known as The Eternal One/The timeless One. The word eternity expresses the concept of something that has no end and/or no beginning. That which is beyond the parameters that we associate with the passage of time. The eternity of the divine mystery is contrasted with the temporality of man who fades into dust.
Celebrity cosmologists such as Stephen Hawking (recently cited on this thread) believe the beginning of the Big Bang was time t = 0, but other cosmologists (including Sean Carroll, Jennifer Chen, and Julian Barbour) believe our universe had a Big Bang event but that the universe did not begin with this event, and that our universe may be eternal in the past and future. Here you may appreciate that there are many conflicting understandings among scientists, when attempting to understand the beginning of time per the reference point of the Big Bang, as an indicator/mile stone marking the "birth" of physical matter.
We can conclude that there was a beginning to life, as we human beings, understand existence...for our scientists tell us so...returning us to the original question...what was responsible for the...beginning of existence?
Were we to conclude that the emergence of physical matter i.e. The Big Bang, came out of nothingness...we are obliged to accept this idea in "faith."
It can be argued, and theists so argue...that the marking of time is irrelevant to the divine mystery, because it transcends time.
I deliberately choose to use the term, divine mystery for the properties of the creator, are beyond human understanding. We may understand the attributes of the divine mystery such as love, forgiveness, and understanding when living in relationship with its nurturing nature...as a parent loves, and shepherds its children.
“From everlasting to everlasting You are God” ~Psalm 90-2