The Typikon, the calendar-handbook of the Church, specifies that there be one liturgy celebrated on the altar per day. If there are multiple priests present, they concelebrate Liturgy, rather than celebrate several liturgies in sequence. (This can be ignored if the building cannot hold all the people in the parish and there is a true need for a second liturgy.)
At Pascha, we celebrate a whole series of liturgical actions leading up to the Matins of the Resurrection and the Divine Liturgy of Pascha. Since we generally do the Matins and Divine Liturgy of Pascha at night on Saturday -generally around midnight--, then the liturgical celebration is finished when the Liturgy is finished. Thus, there's nothing going on at church on "Easter Sunday" until it is time to celebrate the "Vespers of Love" (Agape Vespers). (This should be done in the late afternoon/early evening of Sunday, but a lot of churches do it earlier in the day--sometimes even in the morning hours--to accommodate the domestic celebrations of families. [Gotta get the chocolate bunnies in sometime!]
So, most Churches that follow the customs of the ancient See of Constantinople are empty and locked on "Easter Sunday". Because we did it all--in aces!-- the night before.