This is how the thing works, to open a daily quest from the loyalty steele pillar in Torrent Map.
You must gain a certain amount / quota of loyalty contribution for the daily quest to become available for a particular level ranges.
You can improve the value of the your loyalty by donating rarities to the loyalty steele pillar
[1 set of lvl 6 rarity is equivalent to 1400 loyalty value points and 100 pcs of virtue incense is equivalent to 20K loyalty value points.]
Below is my rough estimate of the values base on the level:
Level 50 - 65 [ Level 4 Rarities ] ( 9K - 11K or higher Loyalty values )
Level 66 - 75 [ Level 5 Rarities ] ( 55K or higher loyalty values )
Level 76 - 95 [ Level 6 Rarities ] ( 90K or higher loyalty values )
Level 96 - 109 [ Level 7 Rarities ] ( 200K or higher loyalty values )
Level 110 - 125 [ Level 8 Rarities ] ( 530k or higher loyalty values )
Level 126 - 139 [ Level 9 Rarities ] ( 730k or higher loyalty values )
Level 140 - 150 [ Level 10 Rarities ] ( 1,250,000 loyalty value )
On achieving the right loyalty values for the loyalty quest to open,
You will be able to gain access / receive the quest name "Heritage Sites" right below the Donate Loyalty from the Loyalty Steele Pillar.
The character are required to complete the 3 main trial quests starting at Level 50, 66 and 76.