Praise be to Allah the Lord of the Worlds.
Tarawih is similar to Qiyam in the sense that both are offered at night and both are recommended acts of Sunnah. However, Tarawih refers to prayer at the beginning of the night, but Qiyam, in general, isn`t considered part of Tarawih.
In addition, Tarawih prayer, in some of its features and conditions, differs from Qiyam. Tarawih refers to Qiyam at the beginning of the night particularly during Ramadan; whereas, Qiyam during other months refers to optional night prayers performed for no specific intention.
Moreover, it is recommended that a Muslim perform Tarawih prayer in congregation while this isn`t recommended in Qiyam prayer. In addition, Tarawih is prayed during Ramadan with the intention of Qiyam or Tarawih. Whereas, intention in Qiyam prayer isn`t specified.
Further, it is recommended that a Muslim offer a certain number of Rakahs in Tarawih. In his Musanaf, Ibn Sahybah narrated from Ibn Abbas that " The Messenger of Allah used to perform twenty Rak`ahs (Tarawih) and Witr prayer during Ramadan. Moreover, in his book {As-Sunnan Al-Kubra}, Al-Baihaqhi narrated from Ibn Abbas that " The Prophet (PBUH) used to perform twenty Rak`ahs (Tarawih) and Witr prayer outside the congregation."
Tarawih prayer is offered in two Rak`ahs unit. Therefore, it isn`t permissible to offer more than two Rak`ahs with one Tasleem. For example, offering four Rak`ahs with one Tasleem renders it invalid as Tarawih because it becomes like obligatory prayers. On the contrary, it is permissible to offer Qiyam as two Rak`ahs or four or whatever number a person likes.
It is better to offer Qiyam at home; whereas, it is better to offer Tarawih at the mosque.
As for Tahajjud, it refers to prayers offered after sleeping. And Allah the Almighty knows best.