How can God be constantly three persons at the same time?
The difficulty comes from a rule we picked up by counting creatures: one being, one person. God is not a creature, so that rule does not apply to Him — and He does not take turns (Father, then Son, then Spirit). All three act together in the same moment, as they do every time a Christian prays: to the Father, through the Son, by the Spirit.
There is an assumption buried in the question, and the assumption is where the difficulty comes from.
Think about how you learned what a person is. You learned it by meeting people. Every one of them arrived the same way — one body, one mind, one voice, one person. So without ever deciding to, you formed a rule: one being, one person — one creature.
God is not a creature. So when Scripture presents one God in three persons, it is not breaking a rule of logic. It is telling us that a pattern we picked up from counting people does not stretch to cover the One who made them. That does not explain how God is three persons at once — nothing explains that. What it does is remove the reason for thinking it impossible, because the reason turned out to be a habit of counting rather than a fact about God.
There is a second assumption often hiding in the question: that God takes turns. Father in the Old Testament, Son in the Gospels, Spirit ever since. This is the natural guess of a quick reader, and Scripture closes it off at every point. The Son did not begin at Bethlehem — He was with the Father before the world existed. The Father did not stop being Father when the Son was born. And the Spirit is already at work in the second verse of the Bible, moving over the face of the waters.
Look at what Scripture says is happening right now, every time a Christian prays.
Ephesians 2:18 (NKJV)
For through Him we both have access by one Spirit to the Father.
One short sentence, three persons, one act. You come to the Father. You come through the Son. You come by the Spirit. Not one after another — all three in the same moment, in the same prayer.
Jesus described the same thing on the night before He died.
John 15:26 (NKJV)
But when the Helper comes, whom I shall send to you from the Father, the Spirit of truth who proceeds from the Father, He will testify of Me.
Count the persons in that sentence. The Son sends. The Spirit comes from the Father. The Spirit testifies about the Son. Three, told apart from one another, in a single breath — and not one of them standing aside while another works.
This is why they are described as distinct but never divided. The Father is not the Son; the Son is not the Spirit. Yet they never act against one another, and never apart from one another. What one wills, all will, because it is the will of the one God.
An ancient summary of the faith, known as the Athanasian Creed, put it in one line: the Father is God, the Son is God, the Holy Spirit is God; and yet there are not three Gods, but one God. That is not a riddle the church invented and then had to defend. It is the shortest way of saying everything the Bible says, all at once.