Random quantum circuits can form approximate unitary designs and pseudorandom unitaries in shallow depths. The paper proves that random circuits on any geometry, including 1D lines, can generate approximate unitary designs over n qubits in log n depth. Similarly, pseudorandom unitaries (PRUs) can be constructed in poly log n depth for both 1D and all-to-all-connected circuits. These shallow circuits have low complexity and create only short-range entanglement but are indistinguishable from unitaries with exponential complexity. The construction glues local random unitaries on log n or poly log n sized qubit patches to form a global random unitary on all n qubits. In the case of designs, local unitaries are drawn from existing approximate unitary k-designs, while for PRUs, they are drawn from ensembles conjectured to form PRUs. Applications include classical shadow tomography with log-depth Clifford circuits, superpolynomial quantum advantage in learning low-complexity systems, and quantum hardness for recognizing topological phases. The results show that shallow circuits can replicate the behavior of Haar-random unitaries, which is crucial for quantum technologies like benchmarking, cryptography, and quantum computing. The paper also discusses the contrast between quantum and classical circuits, showing that quantum circuits can achieve Haar-random behavior in much lower depths due to non-commuting observables. The results have wide-ranging implications for quantum information theory and quantum computing.Random quantum circuits can form approximate unitary designs and pseudorandom unitaries in shallow depths. The paper proves that random circuits on any geometry, including 1D lines, can generate approximate unitary designs over n qubits in log n depth. Similarly, pseudorandom unitaries (PRUs) can be constructed in poly log n depth for both 1D and all-to-all-connected circuits. These shallow circuits have low complexity and create only short-range entanglement but are indistinguishable from unitaries with exponential complexity. The construction glues local random unitaries on log n or poly log n sized qubit patches to form a global random unitary on all n qubits. In the case of designs, local unitaries are drawn from existing approximate unitary k-designs, while for PRUs, they are drawn from ensembles conjectured to form PRUs. Applications include classical shadow tomography with log-depth Clifford circuits, superpolynomial quantum advantage in learning low-complexity systems, and quantum hardness for recognizing topological phases. The results show that shallow circuits can replicate the behavior of Haar-random unitaries, which is crucial for quantum technologies like benchmarking, cryptography, and quantum computing. The paper also discusses the contrast between quantum and classical circuits, showing that quantum circuits can achieve Haar-random behavior in much lower depths due to non-commuting observables. The results have wide-ranging implications for quantum information theory and quantum computing.