They exist. We give names to things we observe, and these are the names we've given to the things we observe. In effect, we've given two names (time and space) to the same phenomenon (spacetime). You might say that spacetime doesn't exist either, but whatever its true nature is, we've named what we observe.
Pretending, for a moment, that you do not travel through time or space, that what only exists is right now, and that there is no direct link between past memories and future memories, space and time still exists as descriptions of perception.
Forgetting quantum entanglement for a minute (because that's stuff is confusing). Let's talk about black holes.
Where is a black hole? That's a confusing question, it's a singularity in spacetime. It's about as confusing as saying when is a black hole. As you travel into a black hole,
time slows down for you relative to the rest of the universe. In front of you is the beginning of the black hole (time is moving so slowly at the center of a black hole compared to you right now, that it just began), and behind you is the end of the universe (time is moving so fast outside of the black hole compared to the inside that all of time transpires while you fall in). So you never hit the bottom (compared to our perception of time). Black holes expose the relativity of the experience of space and time. They exist without space or time, which led me to make
this post about black holes. Where are they? Nowhere, anywhere. When are they? Never, always.
Our perception of time is directly analogous to our perception of space. We perceive time because of where we are, and we are where we are because of our perception of time. We surf the surface of spacetime and hope not to fall off (into a black hole) leaving spacetime altogether.